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LONDON MATHEMATICAL SOCIETY
MONOGRAPHS NEW SERIES
Series Editors
H. G. Dales Peter M. Neumann
LONDON MATHEMATICAL SOCIETY MONOGRAPHS
NEW SERIES
Previous volumes of the LMS Monographs were published by Academic Press,
to whom all enquiries should be addressed. Volumes in the New Series will be
published by Oxford University Press throughout the world.
NEW SERIES
1. Diophantine inequalities R. C. Baker
2. The Schur multiplier Gregory Karpilovsky
3. Existentially closed groups Graham Higman and Elizabeth Scott
4. The asymptotic solution of linear differential systems M. S. P. Eastham
5. The restricted Burnside problem Michael Vaughan-Lee
6. Pluripotential theory Maciej Klimek
7. Free Lie algebras Christophe Reutenauer
8. The restricted Burnside problem {2nd edition) Michael Vaughan-Lee
9. The geometry of topological stability Andrew du Plessis and Terry Wall
10. Spectral decompositions and analytic sheaves J. Eschmeier and M. Putinar
11. An atlas of Brauer characters C. Jansen, K. Lux, R. Parker, and R. Wilson
12. Fundamentals of semigroup theory John H. Howie
13. Area, lattice points, and exponential sums M. N. Huxley
14. Super-real fields H. Garth Dales and W. Hugh Woodin
15. Integrability, self-duality, and twistor theory L. Mason and
N. M. J. Woodhouse
16. Categories of symmetries and infinite-dimensional groups Yu. A. Neretin
17. Interpolation, identification, and sampling Jonathan R. Partington
18. Metric number theory Glyn Harman
Metric Number Theory
Glyn Harman
School of Mathematics
University of Wales Cardiff
CLARENDON PRESS • OXFORD
1998
Oxford University Press, Great Clarendon Street, Oxford 0X2 6DP
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©Glyn Harman, 1998
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Harman, G. (Glyn), 1956-
Metric number theorylGlyn Harman.
(London Mathematical Society monographs; new ser., 18)
Includes bibliographical references (p. - ) and index.
1. Number theory. I. Title. II. Series: London Mathematical
Society monographs; new ser., no. 18.
QA241.H316 1998 512\7—dc21 97-51940
ISBN 0 19 850083 1
Typeset by Newgen Imaging Systems (P) Ltd, Chennai, India
Printed in Great Britain by
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Dedicated to Ruth, Matthew, Jonathan and Christopher
Preface
Topics in metric number theory sometimes appear in books of their own (for
example that of Sprindzuk [254]), and at other times as part of a wider discussion
of a subject (for example uniform distribution [175] or continued fractions [229]).
It is the purpose of this monograph to bring together several different types of
result in metric number theory. Some of these have not appeared in book form
before (in particular the non-periodic problems encountered in Chapters 6, 7, and 8).
Other results included here have appeared only in books now out of print. Since
the metric theory of continued fractions has received treatment recently
(by Rockett and Sziisz [229] and by Schweiger [248]), we shall concentrate mainly
on other aspects of theory. To give a flavour of the difference between our
treatment and that of other recent authors, a typical problem here would be: prove
that for almost all a there are infinitely many prime denominators in the
continued-fraction expansion (or even give an asymptotic formula for the number
of prime denominators not exceeding N asN tends to infinity). A typical question
studied in other works would be: prove an asymptotic formula valid for almost all
a for the number of times a specified digit occurs as a partial quotient in the
continued-fraction expansion.
The general theme permeating this work is the characterization of the number-
theoretic properties of a typical number (by 'typical' we mean all except on a set
of measure zero). We do this first by considering the distribution of digits in
'decimal' expansions to an integer base b, and then by various questions
concerning fractions which approximate a typical number. Some of these questions
are then generalized to higher dimensions—that is, we consider simultaneous
approximation to several real numbers. The most stubborn and important
unsolved problem in this area is the Duffin and Schaeffer conjecture, and we
discuss this in detail. We also consider uniform distribution and properties of
integer sequences which depend on a real parameter (for example [an] for almost
all real a). Our approach will use some elementary combinatorial, measure-
theoretic, and probabilistic ideas together with techniques from analytic number
theory. We occasionally appeal to the language of ergodic theory, but do not
assume any prior knowledge of that subject. The interaction of metric number
theory with ergodic theory and dynamical systems falls outside the scope of
this book.
It goes without saying that this volume reflects the author's own tastes.
Nevertheless, it is the author's intention that this volume should provide an
accessible introduction to the subject for postgraduate students, while giving
viii
Preface
fellow researchers a useful compendium of established results and fundamental
auxiliary lemmas, together with end-of-chapter notes to bring the reader up to
date with current developments, and to suggest further avenues for research. In
addition, parts of Chapters 1, 2, 5, and 10 could form the basis for an advanced
undergraduate/taught postgraduate course in metric number theory.
Cardiff
August 1997
G. H.
Contents
Notation xiii
Introduction xv
1 Normal numbers 1
1.1 Definitions and elementary properties 1
1.2 Metrical lemmas and Borel's theorem 7
1.3 The law of the iterated logarithm 18
Notes 23
2 Diophantine approximation 24
2.1 Statement of results 24
2.2 Zero-one laws 29
2.3 The Duffin and Schaeffer theorem 37
2.4 Vaaler's theorem 44
2.5 Proof of Theorems 2.3 and 2.8 51
2.6 The Duffin and Schaeffer conjecture reformulated 53
Notes 58
3 GCD sums with applications 60
3.1 Statement of results 60
3.2 Proof of Theorem 3.1 67
3.3 Proof of Theorem 3.2 70
3.4 Proof of Theorems 3.3 and 3.4 78
3.5 Proof of Theorem 3.5 82
3.6 Proof of Theorems 3.6 and 3.7 86
3.7 Proof of Theorem 3.8 91
Notes 93
4 Schmidt's method 94
4.1 Statement of results 94
4.2 Proof of Theorems 4.1 and 4.2 97
4.3 Proof of Theorem 4.3 105
4.4 Proof of Theorem 4.4 109
4.5 The metric theory of continued fractions 112
4.6 A generalization to higher dimensions 115
Notes 118
X
Contents
5 Uniform distribution 120
5.1 Definitions and elementary properties 120
5.2 Trigonometric sums, the Erdos-Turan theorem and
the Weyl criterion 126
5.3 The metrical theory of uniform distribution 131
5.4 Uniform distribution in higher dimensions 151
Notes 161
6 Diophantine approximation with
restricted numerator and denominator 164
6.1 Introduction and statement of results 164
6.2 Proof of Theorem 6.2 171
6.3 Proof of Theorem 6.3 175
6.4 Proof of Theorem 6.4 177
6.5 Proof of Theorems 6.5 and 6.6 179
6.6 Proof of Theorem 6.7 180
Notes 186
7 Non-integer sequences 187
7.1 Introduction and statement of results 187
7.2 Proof of Theorems 7.1 and 7.2 192
7.3 A reduction of the problem and
proofs for Theorems 7.3 and 7.5 198
7.4 Proof of Theorem 7.4 202
7.5 Proof of Theorem 7.6 206
7.6 Proof of Theorems 7.7 and 7.8 211
Notes 213
8 The integer parts of sequences 215
8.1 Introduction and statement of results 215
8.2 Proof of Theorem 8.1 220
8.3 Proof of Theorem 8.2 226
8.4 Proof of Theorem 8.3 229
8.5 Proof of Theorem 8.4 233
8.6 Proof of Theorem 8.5 234
8.7 Proof of Theorem 8.6 237
8.8 Proof of Theorem 8.7 238
Notes 240
9 Diophantine approximation on manifolds 241
9.1 Introduction 241
9.2 Proof of Theorem 9.2 245
9.3 Proof of Theorem 9.3 256
Notes 261
Contents
xi
10 Hausdorff dimension of exceptional sets 262
Introduction and statement of results 262
Proof of Theorems 10.1 and 10.2 266
Proof of Theorems 10.3 and 10.4 267
Proof of Theorems 10.5, 10.6, and 10.7 271
Proof of Theorem 10.8 276
Notes 278
280
10.1
10.2
10.3
10.4
10.5
References
Index
Notation
X{ ) represents the Lebesgue measure of a set on the real line, and, where
appropriate, A:-dimensional Lebesgue measure.
li(£), where £ is a set, represents a measure, not necessarily Lebesgue measure.
pi(n), where n is an integer, represents the Mobius function.
(pin) usually represents the Euler totient function.
t(«) denotes the number of divisors of n.
||.x|| indicates distance to the nearest integer.
[x] indicates the integer part of x.
[x] means the nearest integer to x.
{x} represents the fractional part of x.
a.e. means 'almost everywhere',
i.o. means 'infinitely often'.
a ~ A means A^a<2A, except in Chapter 6, where we give the notation a
different meaning.
A s B indicates A = B{\ + o(l)).
Z+ represents the set of positive integers.
%{x) means the number of primes not exceeding x.
A«B indicates A = 0(B).
Introduction
The most natural question to ask at the outset is: what is metric number theory?
The answer we shall assume in this monograph is: the study of the number-
theoretic properties of real numbers or A:-tuples of real numbers from a global
measure-theoretic point of view. We are thus not interested here in the properties
of particular numbers (say e or n), or all numbers of a given class (the set of
algebraic numbers, for example). What will concern us is the problem of finding
which properties hold for almost all numbers (in the sense of Lebesgue measure
on U or Uk). For example, if we write ||x|| for the distance from x to the nearest
integer, and put
f(cc,N)= min ||o/i||,
l<n<N
then
for all real a by Dirichlet's theorem (see Theorem 2.1 below), and this is best
possible (take a = (JV+l)-1 say). If a is irrational, then the theorem of Hurwitz
([141], see [58], Chapter 1, Theorem V) gives
infinitely often. By the celebrated theorem of K.F. Roth [231], however, for every
algebraic irrational and every e > 0,
The above results are all of interest to the number theorist, but the type of theorem
we consider in our present context is as follows.
Let € > 0 be given. Then the following three statements are all true for almost all
real a:
(i) There are infinitely many N such that
/(a,JVK(AaogA0_1.
(ii) For every N,
/(a, A0 ^ C(a)JV~-* (log A0 ~! ~€.
xvi
Introduction
(iii) We have
limsup Nf(oc, N) = \.
N-oo
These results will all be proved in Chapter 2. Sometimes we shall be able to
construct specific examples of numbers belonging to the set of almost all a, or to
the exceptional set of measure zero, but our main interest will lie in finding what
is true for almost all a.
The subject of metric number theory has its origins in the researches of Borel,
Weyl, and Khintchine in the first three decades of the twentieth century. From
Cantor's demonstration that the continuum is uncountable, it follows that 'almost
all' (in the sense that c>X0) numbers are indefinable. This prompted Borel to
enquire into the general properties of 'almost all' (in a measure-theoretic sense)
numbers. The two main contributions made by Borel were in the area of normal
numbers (see Chapter 1) and the growth of the partial quotients in the continued-
fraction expansion of numbers (see Chapter 4 here).
In 1910, Bohl, Sierpinski, and Weyl independently showed that if 0^a^/?<l,
then
"mi Z ! = £-« (0-D
a^{nd}<P
for every irrational 6. Here {x} denotes the fractional part of x. In his monumental
memoir in 1916, Weyl developed this result and gave birth to the study of uniform
distribution. As well as studying specific sequences (such as nkcc for any irrational
a), Weyl also considered metrical problems (such as ana for almost all a). We shall
study these in Chapter 5. It is interesting to note that the original reason for
studying these questions was their relation to problems in applied mathematics
(secular perturbations in astronomy [43] and the statistical mechanics of a gas
[279]). Hardy and Littlewood were also interested in these and related problems
at the time and posed some stimulating questions on the distribution of the
fractional parts of real sequences (see [110] for example). We shall discover that
one problem considered by Hardy and Littlewood—the distribution of {bnoc} for
a fixed integer b—is equivalent to Borel's normal-number result.
In 1923 Khintchine proved a quantitative version of (0.1) for almost all 6, and
this result is essentially best possible. In 1924 he gave a result which implies
statement (i) above, and which has been the starting point for much subsequent
research (Chapter 2 here). Also, in 1924 Khintchine proved a very precise form
of Borel's normal-number theorem, which is the first recorded case of the now
well-known 'law of the iterated logarithm'. We shall prove this result in Chapter 1.
The two subjects of uniform distribution and Diophantine approximation form
the main themes from Chapter 2 onwards in this monograph as we see how
subsequent authors have followed in the footsteps of Weyl and Khintchine. Some
problems are generalized to higher dimensions or submanifolds (see Chapter 9) of IR".
Introduction
xvii
We shall also consider approximations of the form
\na—m\<\j/(n)
where both m and n are restricted to given sets (Chapter 6). New difficulties enter
at this point, since the zero-one law which operates when m is unrestricted (see
Theorem 2.7) is no longer true in general, and the problem is no longer periodic
in a. These difficulties are shared by the problem considered in Chapter 7, where
we study sequences {an<x} where an is an arbitrary increasing sequence of reals,
and also by the questions in Chapter 8 which involve the multiplicative structure
of the integer part of sequences. Finally, we shall touch upon the question of the
Hausdorff dimension of the exceptional sets of measure zero which arise in some
of our earlier theorems. In doing this, we shall generalize results of Jarnik and
Besicovitch from the late 1920s.
When we consider normal numbers in Chapter 1, no background knowledge
of number theory will be presupposed. Treating this simple subject first will allow
us to develop the machinery required for other problems (Lemma 1.5 in particular)
without the complicated technical problems which will arise in the later chapters.
As the book progresses, deeper results from number theory, in particular
multiplicative number theory, will be needed. This should not be suprising since, as is
well-known [98], the distribution of the rationals on the real line is intimately
connected with the Riemann hypothesis. Some of the connections we shall find
with multiplicative number theory arise in all problems of a given type (Chapters
2, 3, and 4), while others (in Chapters 6 and 8) arise because of the type of
sequences we choose to study.
There are many unsolved problems in metric number theory, and we shall
attempt to provide the reader with up-to-date information on the topics as we
proceed. In the author's opinion the greatest challenge in this subject is to prove
the Duffin-Schaeffer conjecture that the divergence of
2, <A(«)—
(where (p(n) is Euler's totient function, and \f/(ri) is non-negative) guarantees
infinitely many solutions to the inequality
\<m—m\ < \f/(ri) with (m, n) = 1
for almost all a.
It is the author's intention that this book should serve as an introduction for
postgraduates to this fascinating area of study, as well as providing seasoned
researchers with a useful compendium of established results and fundamental
auxiliary lemmas. We assume the reader is familiar with the elements of measure
theory (the early chapters of [232] for example) and has to hand a standard text
on number theory (such as [111]). The author must bear the responsibility for any
xviii
Introduction
errors in the proofs. He has sometimes stuck closely to existing proofs, but at other
times introduced what he hopes will be simplifications or clearer arguments.
One topic which features heavily in this work is the estimation of 'overlap
estimates'
m<n<N
where we wish to show that almost all a belong to infinitely many of the sets 31 n.
This leads us to investigate upper bounds for the number of solutions to the
inequality
r s
m n
<e
with various restrictions on the variables.
The reader might be suprised to find that we do not require any Fourier
analysis until Chapter 5: there appear to be no advantages in using Fourier
analysis for the problems discussed in Chapters 1 to 4. Also, the reader should
note that we have been able to use results from the earlier chapters to construct
'bad' sequences in Chapter 5 and so unify certain aspects of the theory (there
appears to be a general principle that one can use good distribution in very short
intervals to produce bad distribution in larger intervals).
Finally, the author would like to thank Professor R.C. Baker, who first
introduced him to this subject via an undergraduate course at Royal Holloway
College, and who gave valuable advice during the writing of this monograph.
Thanks are also due to the anonymous Reader of the first draft, who spotted
several slips and obscurities.
1
Normal numbers
Definitions. Elementary properties. Alternative definitions. Metric
theory. Borel-Cantelli lemmas. Borel's normal number theorem. An
important asymptotic formula (Lemma 1.5). Why should there be zero-
one laws? Law of the iterated logarithm.
1.1 DEFINITIONS AND ELEMENTARY PROPERTIES
Let b represent an integer no less than 2. For any real number a there is a unique
expansion in base b of the form
«=[«] + I «„*"". (l.l.D
n=l
Here [ ] denotes integer part, 0 ^ an < b, and an < b — 1 infinitely often. This second
condition on an is introduced to ensure the unique representation of certain
rationals, of course. For a fixed real number a we write A(d, b, N) to denote the
number of occurrences of the integer d in the set {ax,...,aN} with the an given by
(1.1.1). Following Borel [45], we say that a number a is simply normal to base b if
r A(d,b,N) 1
N^oo N b
for every d with 0 < d < b — 1. We shall call a number entirely normal to base b if it is
simply normal to all bases bn,n = \,2,... This definition differs from Borel's, but we
shall demonstrate below (Theorem 1.2) that our definitions are equivalent. Finally,
we term a real number absolutely normal if it is entirely normal to every base b
greater than 1. As is customary, we shall abbreviate this last expression to normal.
It is of course trivial to exhibit a simply-normal number to any given base (for
example, 0.0123456789 in base 10). The reader will quickly verify that the number
0.12345678910111213..., where the digits are obtained by writing down the positive
integers in order in base 10, is entirely normal to base 10. It will turn out to be
the case that almost all real numbers are normal. We first prove some elementary
properties of normality to given bases. Some properties, such as the periodicity in
a (period 1) of the definition, are self-evident. Others require more work.
2 Normal numbers
Theorem 1.1 Let b and n both be integers no less than 2. Then, if a is simply normal
to base bn, it is simply normal to base b.
Proof Write c = bn. By the hypothesis of the theorem we have
A(d,c,N) 1
= - + o(l) (1.1.2)
N c
for all d with 0< d ^ c — 1. We note that every integer d between 0 and c — 1 can
be expressed in the form
n-l
YjCjWb* with O^cj^b-l.
j=o
Now suppose that
{«}-z^*""=i^_'-
wi=l r=l
Let a be given, with 0 ^ a < b — 1. Then the integer a occurs exactly /: times among
the numbers am with /« + l^w^(/ + l)«if and only if the equation
Cj(dt)=a
has precisely k solutions. Write
D(k) = {d:0^d^c — l, Cj(d)=a has k solutions}.
Clearly D(k) has nCk(b — l)n~k members. Let JVbe a positive integer. We then have
A(a, b, Nn) = " k A(d,c,N)
Nn k = OndeD(k) N
= i-nQ(^-i)n-fc(c-i+oa))
k = On
from (1.1.2)
= "X n-1Cr0-l)n-1-'-(c-1 + o(l))
r=0
= i + o(l)
by the binomial theorem. Because
A(a, b,N+r)~ A(a, b, N) < r,
this gives
r A(a,b,N) 1
lim v = I-
Since this holds for each a, we conclude that a is simply normal to base b. □
Definitions and elementary properties 3
Corollary The statements 'a is entirely normal to base b' and 'a is entirely normal
to base bn' are equivalent.
Proof If a is entirely normal to base b, it is simply normal to all bases bnr, and
so is entirely normal to base bn. On the other hand, if a is entirely normal to base
bn, it is simply normal to base bnt for any t > 1, and so it is simply normal to base
bx by the theorem (replacing b by bf). Hence a is entirely normal to base b. □
We remark that the implication of Theorem 1.1 cannot be reversed. For
example, 0.012345678^ is simply normal to base 10 but not normal to base 100.
We shall use Theorem 1.1 to show that our definition of an entirely normal number
is equivalent to that given by Borel. He calls a number entierement normal if each
of a, oib, ocb2,... is simply normal to every base b, b2, b3,... Clearly, if a number is
entierement normal, then it is entirely normal by our definition. The next theorem
was first demonstrated by Pillai [218], [219].
Theorem 1.2 If a. is entirely normal to base b, then each of cc, ccb, ccb2,... is entirely
normal to every base b, b2, b3,...
Proof In view of the previous corollary by induction, we need only show that ocb
is entirely normal to base b. We shall write A*(d, c, N) to denote the number of
occurrences of digit d among the numbers al9 ...,aN where
oo
boc, = [6a] + Yj ajc~j->
and reserve A(d,c,N) for the corresponding function for a. Since
\A*(d,b,N) —A(d,b,N)\ <1, it is trivial that ocb is simply normal to base b. We
shall prove that ocb is simply normal to every base bj (y'^2) by using the fact that
a is simply normal to every base bjr with r arbitrarily large.
Now let a be an integer between 0 and bJ—1, and suppose that r is a positive
integer. Given an integer d with 0^d^bjr—\, define gm=gm(d) by
bd-ldb1 ~jrWr= '£ bm*gm, 0^gm<b>.
m = 0
We define D(k) to be the set
{d: 0 < d < bjr — 1, gm=a has exactly k solutions for m ^ 1}.
We note that the cardinality of D(k) is
r-1ckbj(bj-\y-k-1.
It follows from the definition of D(k) that
A^b^Nr)^ k X A(d,bJr,N).
k = 0 deD(k)
4 Normal numbers
Hence, since a is simply normal to base bjr,
A*{a,b\Nr) ^ '£k r-ic y(y_lr>-i(^-Jr+o(1))
Nr k% r
= (\-\/r)b-j(\+o(\)).
Thus for any integer M we have
A*(a,bJ,M)
w
It follows that
Xl-r-^b-'a + od)).
. A*(a,bj,N) . _
hmmf — ^b J(l—r l). (1.1.3)
N-oo A'
Since this holds for every a and
"-1 A*(a,bj,N)
a = 0
we must also have
y v ' ' y = 1
limsup l ' ' ; < ZTJ'(1 +Mr). (1.1.4)
JV-oo iv
Because (1.1.3) and (1.1.4) hold for arbitrarily large r, we may deduce that
lim — = b J, for O^a^bJ — 1.
JV-oo N
We have thus proved that ocb is simply normal to every base bj and this completes
the proof. □
There is another common definition of normality, given for example by Hardy
and Wright [111], Niven and Zuckerman [209], and Kuipers and Niederreiter
[175] (see also page 261 of Borel's paper [45]). Given a real number a and a base
b, we express a in the form (1.1.1). Given a block Bk of k digits between 0 and
b — \, we then write A(Bk,b,N) for the number of occurrences of Bk in the block
of digits a1...aN. Some authors then call a number normal to base b if
lim A(B>»b>N) = b-k for all k ^ j and all B
JV-oo N
The next theorem shows that this is equivalent to the definition we have chosen
for an entirely normal number.
Definitions and elementary properties 5
Theorem 1.3 A real number cc is entirely normal to the base b if and only if
lim A(Bk,b,M) = b_k for ^ k^^ and ^ (J t 5)
N-oo A'
Proo/* Suppose that a is entirely normal to base b and let Bk be given. From
Theorem 1.2 we know that each of oc,boc,...,bk~1oc is simply normal to base bk.
Now we can write Bk as a single digit, say d, to base bk. The occurrence of the
block Bk as
for some j with 0 <y < k — 1, is then equivalent to the digit d being in the (r — l)th
position in the expansion of bJa to base bk. Write Aj(d, bk, N) for the number of
occurrences of d among the first N digits in the expansion of bja to base bk. Then
A{Bk,b,Nk) ^X'-'Ajjd^^N)
Nk k £Q N
= b-k(\ + o(\)),
since bJoc is simply normal to base bk. Thus
N-oo
N
and this establishes the 'only if part of the proof.
To prove the more difficult 'if part, we require the following lemma, which
will be used later to prove Borel's normal-number theorem.
Lemma 1.1 Let d be a given digit to base k, and suppose that € > 0. Then the
number of different blocks H of u digits (0 <u<k — 1) for which
deH K
^€W (1.1.6)
is no more than tku for u>u0{e).
Proof First consider those blocks H for which
K deH
The number of such blocks is
X "Cj{k-\y-J=Y, say,
0^j<(u/k)-€u
by an elementary combinatorial argument. Now
u
X uCj(k-l)u-j=ku.
j=o
6
Normal numbers
On the other hand, we have
uCj(br-\y-j =(y+i)(y-i)
«ci+1(^-i)---'-1 u-j-i
_(br-\)(ub-r-e + \)
u + e-ub~r
(putting j = ub~r—B)
<(l-6r(0-l)/w)<(l-e&73)
if 6 > €w/2 and u is sufficiently large. It follows that
Y< (1 -ebr/3ful2ku < ^- (1.1.7)
for all large w. The same bound may be obtained for those H with
^ 1 M
and this completes the proof. □
We now complete the proof of Theorem 1.3 following Cassels [57]. Let d be
a digit to base br for some r> 1 (the case r = \ is trivial of course). We want to
establish that (1.1.5) implies the relation
hm = —. (1.1.8)
N-oo N b'
We can write d as a block of B digits to base b, say B = bl...br. Given two blocks
of digits E, F to base 6 with F no shorter than E, we write /fc • (2s, F) for the number
of times that E occurs in F with the first digit of E appearing in position
h =j (mod r) in F. Now let & = @(s, e) be the set of all blocks H of s digits to base
b such that
max
j
Kj^H)-±
^€5.
By Lemma 1.1 with u = [s/r], H replaced by each of H, H/b, ...,H/br~1 (where
hlbx indicates that the last / digits of H are omitted), we can conclude that @ has
no more than tbs members for all large s>s0, say. Now fix s=max(.s0,r€~1).
Now, for every H e@, we have
A(H,b,N)=nb-s + o(N), by (1.1.5).
Hence
X A(H,b,N)^€bs(nb-s+o(N))^2€N (1.1.9)
for all large N.
Metrical lemmas and Borel's theorem
We now write Dt for the block of digits at ...at+s^1 in the expansion of a in
the form (1.1.1). Let
S(N) = {t:Dt4£, t^N-s+l}.
Then \S(N)\ ^2aN by (1.1.9) for all large N. Suppose that N is a multiple of r.
Then
{s-r + l)A(d,br,N/r)- £ *2-r(*,A)
t=i
^2s:
(1.1.10)
To obtain this, note that each solution to
b1 = aJ, b2 = aj+1, ..., br=ai+r_x
s^j^N-j,j = \ (modr)
contributes exactly s—r + 1 to both expressions whose difference is taken on the
left-hand side of (1.1.10).
For the no more than 2 c N values of t e S(N), we use the trivial bound
0^Rj(B,Dt)^s-r + \.
For t$S(N) we have
RA^Dt)--
^€S.
Hence, by (1.1.10),
A(d,br,N/r)-^-r
^4eN+
2s:
N
+ —
-1
s—r + 1 rbr\s—r + 1
for all large N. Since € was arbitrary, this establishes (1.1.8) as required.
1.2 METRICAL LEMMAS AND BOREL'S THEOREM
□
In this section we shall prove some results which will be used several times in
subsequent chapters. Their present application is to prove Borel's normal-number
theorem (and even a quantitative version of his result). This may be stated as
follows.
Theorem 1.4 Almost all real numbers are absolutely normal.
We need only show that, given any integer b ^ 2, almost all numbers are simply
normal to base b. To see this, write £(b) for the set of reals not simply normal to
base b. Then the set of reals which are not absolutely normal is precisely
0 wo-
6=2
8 Normal numbers
This is a countable union of sets of measure zero, and so has measure zero itself,
as required to establish the theorem. □
We begin with a very simple result which will be applied in the more
complicated lemmas that follow.
Lemma 1.2 (The first Borel-Cantelli lemma) Let X be a measure space with
measure jx. Let Ai (j=1,2,...) be a collection of measurable subsets of X. Then, if
oo
£ n(Aj) converges, (1.2.1)
j=i
almost all members of X (with respect to jx) belong to only finitely many of the Ay
Proof We first note that the subset of X, say £, belonging to infinitely many of
the Aj may be expressed as
oo oo
*= n u ^.
n=1j=n
This is written
<f = limsup Aj
by some authors. Of course, £ is a measurable set, since it is formed by a countable
intersection of a countable union of measurable sets.
We have
oc
t*w < m U Aj) for everv n >J-
• J = n
Hence
oo
Vl{&) ^ X ^Aj) for everv n > J- O-2-2)
j = n
Since (1.2.1) holds, we may make the right-hand side of (1.2.2) arbitrarily small
by taking n sufficiently large. Thus fi(&) = 0, as required.
First proof of Theorem 1.4 Let d be an integer between 0 and b — 1, and, for e > 0,
write @(N,€) for the set of all blocks HofN digits (between 0 and b — 1) such that
deH °
^eiV.
By (1.1.7) we have
\@(N,c)\«bNe-pN
Metrical lemmas and Borel's theorem
for some p = p(z)> 0. Now let
X(N,€)= |J
HeB(JV)
H H+V
bN> bN
where H is here regarded as a number written in scale b. We then have
£ MW,€))« £ e-'*«l.
N=l N=l
Let £(€) be the subset of [0,1) belonging to infinitely many of the X(N, e). Then,
by Lemma 1.2, we have p(E(t)) = 0. Hence the set
*=(J£(2-')
has measure zero. Now, if a is not a normal number, then there is some € > 0 for
which
Aid, b, N) 1
N b
> €
for infinitely many N. Hence a belongs to E(2~r) for some r, and thus to S. Since
<f has measure zero, we conclude that almost all ae[0,1) (and hence almost all
real a) are normal. □
We prove the next lemma mainly for historical interest and to show the
development in technique leading up to Lemma 1.5. With one exception (in section
1.3), we shall always appeal to the stronger results which follow.
Lemma 1.3 (The second Borel-Cantelli lemma) Let X be a measure space with
measure \i, and suppose that fi(X) = T<co. Let Aj (y = l,2,...) be a collection of
measurable subsets of X such that
Tn(AjnAk) = n(Aj)n(Ak) for j^k. (1.2.3)
Then, if
oo
X fi(Aj) diverges, (1.2.4)
almost all members of X belong to infinitely many of the Ay
Proof Without loss of generality we may assume that T= 1. The condition (1.2.3)
is then what is normally called 'independence'. Now let Xj(x) be the characteristic
function of As and write
10
Normal numbers
We have
and, for j ^ k,
It follows that
L-
(x)dfi = fi(Aj)
Xj(x)xk(x) dfx = n(Aj)n(Ak).
' urt^V-^w-w*
Thus, in view of (1.2.4),
Hence, by Fatou's lemma,
(i7-i/«<4»:
N \-l
lim
N-oo
fN(x)2dfi = 0.
This gives
liminf fN(x) = 0 a.e.
N-oo
llmsuP^ 77\ = 1 ae-
and so, since XXjC*) is non-decreasing for each x,
N
Ex/*)-*00 ae-
This completes the proof.
(1.2.5)
□
We shall now use a technique of Weyl [280] to prove a much stronger version
of Lemma 1.3, which will lead to our second proof of Theorem 1.4.
Lemma 1.4 Suppose that X is a measure space with measure \i, and that
0 < fi(X) < oo. Let fk(x) (k = 1,2...) be a sequence of non-negative \i-measurable
functions, and let fk, (pk be sequences of real numbers such that
0</*<<P*<l (£ = 1,2...).
(1.2.6)
Metrical lemmas and BoreI"s theorem 11
Suppose that for every n^\,
' ( I (fk(x)-fk))2d^K J] <pk (1.2.7)
= K®(N), say,
where K is an absolute constant and <S>(N)^> oo as N-*co. Then, for every € > 0,
and for almost all xeX,
I /*(*)= I fk + oU2/3(N)\ogmN) + 2)^+\ (1.2.8)
Remark Note that this result has both a weaker hypothesis and a stronger
conclusion than the previous result. In statistical terms, fk is the mean of fk(x) and
(1.2.7) deals with the variance.
Proof Assume without loss of generality that n(x) = 1. Write
V{N,x)= X fk(x), V(N) = £ fk{x),
E(N,x) = x¥(N,x)-x¥(N).
We now outline Weyl's idea before presenting the details. We pick a sequence
Nl9N2,... and consider the measure of the sets on which E(Nj,x) is 'large'. If
'large' is chosen suitably, and the Nj are spaced far enough apart, the sum of
measures converges. We can then apply the first Borel-Cantelli lemma to deduce
that E(Nj,x) is not 'large' for almost all x. If we write Ej for this 'large' error, we
then have
V(N, x) = V(N) + 0(Ej + Ej_! + *¥(Nj) - V(Nj_,)) (1.2.9)
for almost all x, Nj_1^N^:Nj. We must then balance the requirements that a
sum of measures converges, and the Ej and *F(JVj) —¥(#,_!) terms must not
dominate the main term in (1.2.9).
In practice we take Nj as the smallest integer with
0>(iVJ.)>/(log2;)1+€, y = l,2... (1.2.10)
We then let A-} be the subset of those x e X for which
\E{Npx)\>j\\oglj)l+\
By (1.2.7) we then have
ed^Kj\\og2j)l+' + \ „
W* Mogljf^ <<J (1°g2y) *
12 Normal numbers
Hence
oo
converges. From Lemma 1.2 we deduce that, for almost all xeX, there is an
integer t(x) such that x$Aj for ally ^ t(x). Now we have, by (1.2.6), that
^(^-^(Arj._1)^0(^)-<E)(^-1)<<y2(log2y)1+£.
Also, by (1.2.10) with (1.2.6),
y2(log 2J)1 +€«<KNj- ^'WogmNj.,) + 2))(1 +€>'3.
Putting this information into (1.2.9) yields that, for almost all x, there is a t(x)
such that, for N ^ Nt{x),
V(N, x) = TOO + <9(0>(A02/3 \og(p(N) + 2)1/3 +€).
This completes the proof. □
We can now give our second proof that almost all real numbers are simply
normal to a given base b. As before, we need only restrict our attention to [0,1).
By the Mi digit of a real xe[0,l) we shall mean the integer ak given by (1.1.1)
with a=x. Let d be an integer with 0 < d < b — 1 and write
f 1 if the A;th digit of x is d,
Aw-j0 otherwisej
We then have, for j^k,
fk(x)fj(x)dx = n({xe[0,\):kth andyth digits of x both d}) (1.2.11)
= b
-2
This last equality, which the reader may regard as self-evident, follows by noting
that the set in (1.2.11) is the union of certain intervals (supposing k ^y) of the type
[db-j+db-k + oi, db-j+b-k(d + \) + oi),
where a ranges over all fractions of the form
Y,anb~n, 0^an^b-\.
n=l
There are bk~2 intervals each of length b~k, and this gives the result.
It follows that
fl / N
'1 / N \ 2 N
0 \k=l J k=l
Metrical lemmas and Borel's theorem 13
(the variance of fk(x)—fk is &_1(1 — b'1)). We may thus apply Lemma 1.4 with
q>k = b~1, K = \, and so obtain
A(d,b,N) = X /*(*)=£ + 0(N2'3(\og(N+2)))
for almost all xe [0,1) (taking e = 2/3). Thus
r A(d,b,N) 1
lim 5^ = I
for almost all a, and this completes the proof. Indeed, we obtain
We now give a refined version of Lemma 1.4. The result will be used several
times in the subsequent chapters. This lemma appears with a slightly stronger
hypothesis in [254]. The version we give is actually required in [254] because the
condition q>k ^ 1 (which we dispense with) is not always satisfied in Sprindzuk's
application of the result. The development of this important tool in the metric
theory of numbers may be traced through the works of Rademacher [226], Gal
and Koksma [102], Cassels [55], W.M. Schmidt [240], and W. Philipp [213].
Lemma 1.5 Let X be a measure space with measure \i such that 0 < u(X) < oo.
Let fk(x) (k = 1,2,...) be a sequence of non-negative u-measurable functions, and let
fk, (pk be sequences of real numbers such that
0^fk^<Pk (k = 1,2,...). (1.2.12)
Write
«>(A0= I <Pk,
k=l
and suppose that ®(iV)->-oo as JV->-oo. Suppose that for arbitrary integers m,n
(\^m<ri) we have
I (fk(x)-fk))2du<:K X cpk (1.2.13)
for an absolute constant K. Then for any given € > 0, and for almost all x, we have,
as N^-co,
£ /*(*)= I A+ofa>1/2(iV)(log(<D(iV) + 2))3/2+€+ max f\ (1.2.14)
Proof The most inefficient part of the last lemma was the use of (1.2.9), which
suffers from the wide spacing of the Nj there. Here we fill up the gaps in the
14 Normal numbers
spacing of the Nj by using (1.2.13) and a more subtle argument. Let the sequence
nl9n2,... be defined by
rij =max{«: <£(«) <j}.
We note that the rij need not be distinct. For the moment, suppose that (1.2.14)
holds for N=rij for all j. Then, if nr < n < nr+19 we have
£ A(*K I A( Wl A(*X
fc=i fc=i fc=i
while
I A(*) = I A + 0(r1/2(log(r+2))3/2+€),
fc=l Jlc=l
and
"l AW = "l A + 0((r+ l)1/2(log(r+3))3/2+€).
fc=l Jlc=l
Also, by (1.2.12),
X /t^ max/k + <D(«r+1)-<D(«r+l)
^l+max/k.
Combining these results then gives
I A(*) = I /Jk+0(r1/2(log(r + 2))3'2+* + max/k),
fc=l fc=l fc<"
which establishes (1.2.14). We can therefore concentrate on proving the formula
for N=rij.
We begin by expressing the integer j in binary scale as
We then write
B(j) = {(i,s):i= £ bU,v)2v-;bU,s) = \,0^s^r}9
v = s+l
where r = r(j) = [\og2j]. Also we define F(i,s,x) by
F(i,s,x)= £ (AC*)-/*),
u0<fc^u1
Metrical lemmas and Borel's theorem 15
where
ut = ut(i,s) = msix{n>0:<S>(n) <(i + t)2s}, (1.2.15)
with the convention that max 0 = 0. The purpose of this notation is to split up
[l,»j] into a suitably small number of blocks. We have
(0,h,.]= |J (u0,Ul]
(i.s)eBC/)
with u0,ul given by (1.2.15), where the union is disjoint. For example,
5(37) = {(0,5), (8,2), (36,0)} as we go 0 to 32 to 36 to 37 in the binary
representation of the number. Clearly B(j) has no more than r(j) members.
In view of the above, for every x,
I/*(*)= E/* + Z F{i,s,x).
fc=l k=l (i,s)eB(j)
To establish (1.2.14) for N=rij and so complete the proof, it therefore remains to
demonstrate that
X \F(i,s,x)\=OUll2(\ogU + 2))3/2+') (1-2.16)
(i,s)eBU)
for almost all x.
We put
G(r,x)= £ F2(i,s,x),
OagssSr
i<2r"'
0(I,5)= £ <pk,
u0<k^.ul
with ut given by (1.2.15). Then, from (1.2.13), we obtain
r
G(r,x)dfi^K X $('»
0^i<2r_5
^ K(r + \)<!>(n2r)<K(r+\)2r.
Hence
00 00
X fi({xEX:G{r,x)>r2+€2r})<2Kfi(X)YJ r'1'*
r=l r=l
< 00.
We can therefore apply Lemma 1.2 to conclude that
G(r,x)<r2+*2r (1.2.17)
when r > r(x), for almost all x.
16
Normal numbers
Now let r=[log2y] + l, and suppose x belongs to the set for which (1.2.17)
holds. Then an application of the Cauchy-Schwarz inequality yields
X \F{i,s,x)\*k\BU)\1,2Gli2{r9x)
U.s)eB(j)
<r1/2G1/2(r,x).
This establishes (1.2.16) in view of (1.2.17) and the proof is finished. □
It will become apparent in the next section that the exponent 1/2 attached to
O(A0 in (1.2.14) is best possible. The logarithmic factor cannot be replaced by a
factor smaller than (loglog®(iV))1/2. It is not known what is the slowest-growing
function which always suffices in our later applications of the lemma to non-
independent fk(x). Of course, Lemma 1.4 quickly leads to the result that, for
almost all a,
A(d,b,N)=Nb-1 + 0(Nll2(\og2N)3l2+€).
We conclude this section with a brief discussion on why the results we consider
in this book hold for 'almost all x' or 'almost no x\ with no half-way house (except
in certain artificial situations). We make the dependence of A(d, b, N) on x explicit
by writing Ax(d, b, N). Let
f = <xe[0,\): hm = -
I N-oo N b.
and suppose that /*(</) = € > 0. We write
af + P = {xa + fS:xef}.
Now let J be any subinterval of [0,1) of length y. Then there are integers r,s
such that
fb~r+sb~r<^J, b~r^y^b
-r+2
Write Jfr = fb-r+sb~r. Thus \i{X) = ^b~r. If yeX, then y=xb-r+sb~r for
some xef. Hence
Ay(d,b,N) Ax(d,b,N) + 0(r) 1
— = ► — as jN—>oo.
N N b
Therefore X^/. We have thus proved that \i(Jc\#) >\i{J)b~2^. At first sight
this seems to be a rather weak statement. It turns out to be the case, however,
that this establishes that \i(#) = 1, since almost all points of a measurable set have
metric density 1. We prove this in detail in the following result, which dates back
at least as far as the work of Knopp [161].
Lemma 1.6 Let stf be a given subset of U. such that
A{dnJ)^5X(J)
(1.2.18)
Metrical lemmas and Borel's theorem
17
for every finite interval J c R, where b is some positive constant. Then almost all
real x belong to $0.
Proof We first recall Lebesgue's density theorem:
For almost all points x of a measurable set $ we have
hm =1. (1.2.19)
€-0 2€
This result is just a consequence of the result
d rx
dx
Ay) dy =/(*) for almost all x
o
for non-negative measurable functions /.
We prove the lemma by contradiction. Write (£ = U\<b/ and suppose that
A(#) > 0. Then, by (1.2.19), there exists an xe<£ and an € > 0 such that
A((x-e,;c + €)n#)^2€(l-<5/2).
Thus for an interval # we have
U/nV) ^ A(/)(l -5/2). (1.2.20)
Since # = U\ji/, we have
A(/) = A(j/n/) + A(<ifn/)^(l + 5/2U(/)
from (1.2.18) and (1.2.20). This is an absurdity, since A(/)>0. Hence A(#) = 0
and the conclusion of our lemma holds. □
It follows from Lemma 1.6 and the argument which preceded it that the set of
simply normal numbers to a given base has measure zero or one. We note that
Lemma 1.6 says something about the structure of Lebesgue measurable sets.
Roughly speaking, if a set is fairly evenly spread out, then either it or its
complement has measure zero. From a number-theoretic point of view, one
segment of the real line is much the same as any other, once we fix one interval
and investigate the properties of the numbers within it. Differences may arise over
whether fractions with small denominators lie in the interval, and clearly the ratios
of numerator to denominator will vary from interval to interval, but we shall see
that these cause no serious problems.
The above argument can be viewed another way. If we let / be the subset of [0,1)
consisting of numbers simply normal to base b, and write Trs for the mapping
given by
xTr!l = xb~r+sb~r (mod 1),
we then have
In other words, £ is an invariant set under Trs. If we prove that Trs is ergodic
(the only invariant sets have measure zero or their complements have measure
18
Normal numbers
zero), then # has measure 0 or 1. This approach is used in Section 2.2 dealing
with a different problem. This is a parallel approach to the above, especially since
the proof that Trs is ergodic uses the Lebesgue density theorem. Alternatively, one
can prove that almost all numbers are normal and deduce that Trs is ergodic!
A third way of dealing with these problems is from a statistical direction. If
the events we are considering were independent, then the probability of infinitely
many occurring would be 0 or 1 by Kolmogorov's theorem (Theorem 4.5 in
Billingsley [41], for example). The problem usually arises that the events we
investigate are not independent. Of course, in the normal number problem we do
have independent events, since the probability that the nth digit is d is independent
of the value of the rath digit for m#n. Lemma 1.4 demonstrates however that a
suitable bound on the 'variance'
Z (fk(x)-fk))2dpi
X\m^k<n
is sufficient to prove that
Jlc=l Jlc=l
for almost all x.
1.3 THE LAW OF THE ITERATED LOGARITHM
Readers with a background in probability will have already deduced the following
result from the independence of the functions fk(x) and fj(x). We shall give a
self-contained proof for the case b = 2 (which the reader may easily modify for
b > 2), assuming no background in probability. We write
V(b,d,N) = A(d,b,N)-Nb~1.
Theorem 1.5 Let beZ, b^2. Then, for almost all real numbers a,
V{b,d,N)
li^Pa(27Vloglog7V)^2=1'
(1.3.1)
and
V(b,d,N)
liminf- ^ , , .,-
n-oo a(2AnoglogA01/2
= -1,
(1.3.2)
-i>
wherea2 = b-\\-b-1).
Remarks This result was first proved by Khintchine [151], and is the first
occurrence of the 'law of the iterated logarithm' in the literature. A survey of this
phenomenon has been given by Bingham [42]. From (1.3.1) and (1.3.2) we know
a great deal about the distribution of the digits to base b to a 'typical' real number.
Essentially, A(d,b,N) rarely exceeds Nib by more than er(2./VloglogJV)1/2, but
The law of the iterated logarithm 19
infinitely often the excess comes close to this amount. A similar statement holds
for A(d, b, N) being exceeded by Nib. In essence the proof hinges on most of the
contribution to the binomial expansion of (i+i)n coming from the terms nCr2~n
with r close to nil.
Proof We shall treat only the case b = 2 for simplicity. We also establish only
(1.3.1); the proof of (1.3.2) follows immediately in the case b = 2 (and can be proved
similarly for b ^ 3). As before, we restrict ourselves to ae [0,1). We prove (1.3.1)
in two stages. First we establish that
limsup ————;—r^TTi^l ae- (1.3.3)
n-oo <7(2A^loglogA^)1/2
From this we instantly get
. c V(b,d,N)
hminf —;——-772 >—i a.e.,
n-oo <7(2iVloglogA01/2
which we shall need to obtain
limsup _„.—;——-T75 ^ 1 a.e.
n-oo a(2A^loglogA^)1/2
to complete the proof of (1.3.1).
Since we assume b = 2, we write A(d,N) for A(d,b,N), and V(d,N) for
V(b,d,N). We shall often suppress the dependence on d in the following. We put,
for a > 0,
J (a, N) = {a € [0,1): V(d, N) ^ aNl/2},
/(a,N) = {ae [0,1): maxV(d9n) ^aN1/2},
ns£N
^(/i,a) = {ae[0,l): max V(d,m)<aN1/2,V(d,n)^aNl/2}
m^n— 1
(we pause to note that /(a, N) is the disjoint union of the s/(n, a) for n ^ N),
@(n,N) = {<xe[0,\):V(d,N)^V(d,n)}.
We note that, by the independence of the fj(x) (see (1.2.11)), we have
;.(#(«,W)) = i for n<N. (1.3.4)
Before proceeding with the proof, we require the following lemma.
Lemma 1.7 We have, for all large N, and all ae[\, (logJV)1/4],
e-2az
—— ^AG/(a,A0)^e-2fl\ (1.3.5)
4a
20 Normal numbers
Proof We shall assume that N is even, say N=2M, so that
X(J(a,N))=\- X 2MCM+k2~2M, (1-3.6)
where g=aN112 <-|M. By applying Stirling's formula, namely
«!=«n+(1/2)e_n(27r)1/2(l + 0(l/«)), as /i-*oo, (1.3.7)
to a binomial coefficient we find that
1-tW) 1 + ^J (l-£) (1 + 0(1/M)).
(ttM)1/2V m2; \ m) V ^
i
'(TtM)
J72 exp(- A:2/M)(l + 0(l/N+k3/N2))
= TTFJii exP(" k2IM)(\+0{5)), (1.3.8)
with S=N~l/5. On writing 9 = y/2a = gM~a/2) and considering
-^ X exp(-A:2/M)
1V1 0*Zk<g
as a Riemann sum, we see that its value differs by at most <5 from the value of
the integral
re 1/2
exp(-y2)dy=-—-im,
Jo z
with
r«>
m=
exp(-j>2)dj>
From (1.3.6) and (1.3.8) we now find that
It remains to estimate 1(6), which has the asymptotic value exp( — 02)/(20) as
0->oo. In fact, from integration by parts, it is easy to verify that
2eXp(~^</(0)<iexp(-02), ford^Ji,
50
so that the required result (1.3.5) follows. □
The law of the iterated logarithm 21
We now consider ?.(/(a,N)) for
AT=2[(1+ e)fc], a = (l + €)(iloglogiVr)1/2.
We have
N
A(f(a,N))=YJ^(n,a)).
n = g
Also, S(a, N) is contained in
N
(J jrf(n,a)n@(n,N),
n = g
and
u^t a at am \H^(n,a))H<%(n,N)) for n<N
\A(jrf(n,a)) for n = N.
Hence
N
X{Jf{a,N))> X iA(j/(n,fl))=iA(/(fl,JV)).
n = g
Thus, by Lemma 1.7,
2(/(fl,iV))^2e-2a2<—-4ttt7« J
l+c ^M+e
(logA01+e cfc
by our choices for a and JV. Hence, the set on which V(d,N)^aN112 has infinitely
many solutions has measure zero by Lemma 1.2. Let a belong to the set on which
there are only finitely many solutions and suppose V(d, N) ^ aN112 for N ^ N0.
Letn>N0. Then #*_!<« ^JVfc with JV,. = 2 [(1+ €)']. WhtcN=Nk. Thus
V(d9 n) < V(d, N) (NloglogNV12
(nloglog«)1/2 ^ (AHoglogiV)1'2 \ wloglogw /
N V(d9 N) (l + €)2
n (AHoglogAO1'2 21/2 '
We have thus established that
r V<<b,d,N) w1 , x2
limSUP /UM 1 ^1/2 ^ 0 + €) a-e-
n-oc (iAMoglogAO '
Since € was arbitrary, this establishes (1.3.3).
Now let Nk = e~k, Mk = Nk-Nk.v For k ^ 1, write
D(*) = K(rf,J\r4)-F(</, #*_!).
Let
AA:) = {a€ [0,1): i)(A:) ^ ((1-€)iMfcloglogMfc)1/2}.
22 Normal numbers
By the independence of /,(*) and the definition of D(k), we have
k{£{k)n£{j)) = X{£(1c))k{6{j)) for j*k.
Then, by the argument of Lemma 1.7, and noting that
A (*(*)) = KS(Mk9 (1 -€)1'2(iloglog ilf^)1'2)),
we obtain
W(k)) ^iexp(-(l-€)loglogMfc)
1 1 1
>
4(logMk)1_€ 4(logJSTJk)1_< 4A:1_e(-log€)1-€
Hence the series
oo
I Km))
fc=i
is divergent, so almost all a belong to infinitely many of the £(k), by Lemma 1.3.
Let a belong to infinitely many of the £(k) and also to the set of full measure for
which
. V{d,N)
limmf /um i TKui > ~J •
n-oo (iAHoglogAO1'2
It follows that, for a € <f (A:),
F(rf,JVk) D(k) , VidtNt-J
1/2 +
(JVk loglog JVk)1/2 (iVk loglog iVk)1/2 (JVk loglog iVk)1/2
> /(l-€)Mk loglog MkV'2 _ e1'2!^,^)!
^ 2iVkloglogiVfc ) (A^loglogJVJ1'2
^2-(1/2)(1_6)3/2_ ^W^-*)!
(iVk_, loglog N,.,)112'
Hence
limsuP /vi^f'^vM/i *2_a/2)((1 "€)3/2 "€l/2)-
fc-oo (A^k loglog ATk)1/2
Since € was arbitrary, this establishes
limsup ——^ ^1
n-oo (^N loglog AO '
and so completes the proof of Theorem 1.5. □
Notes
23
NOTES
For the theory of other representations of a number, see the monograph of
Galambos [103]. The number 0.1234567891011121314... is known as
Champernowne's number [62]. There has been much interest in showing that
numbers 0.a1a2a3..., where the a} are blocks of digits to base b corresponding to
an increasing sequence, are entirely normal to base b. Examples include a- as the
yth prime [66], or aj=f(j), where / is a non-constant polynomial taking only
positive integer values [68]. Some of these constructions still yield normal numbers
to base b when the blocks are perturbed slightly [273]. Many authors have given
constructions of absolutely normal numbers (for example [177], [183], [238]).
Others have carried out numerical tests on the fundamental constants of
mathematics, such as e or 7i.
One family of problems in this area relates to studying numbers entirely
normal to one base (or set of bases), but not to another base (or set of bases). By
Theorem 1.1, if (logr)/(logs) is rational, then normality to base r is equivalent to
normality to base s. If (logr)/(log.s) is irrational, the set of numbers normal to base
r, but not to base s, has been shown to be uncountable [237], and even to have
dimension 1 [202]. This result has since been extended to cover numbers normal
to one set of bases SP, but not another 01', provided that re@, seSf implies that
(logr)/(logs) is irrational [223]. A different point of departure for research is to
consider representations to non-integer bases [198]. We shall return to some of
these questions in Chapters 5 and 10.
The fact that almost all numbers are simply normal to any given base is a
consequence of the individual ergodic theorem [40], as first pointed out by Riesz
[228].
A weaker condition than normality for a number is that every block of digits
to base b should occur infinitely often. This problem has also aroused some
interest. Mahler [189] proved that if a is irrational, then there is an integer
X ^X0(N,b) such that each block of length N occurs infinitely often in the
expansion of Xa to base b (see also [261]).
Other questions regarding patterns of digits in blocks will be discussed in
Chapter 2 (iong' blocks of zeros), Chapter 5 (where the equivalence of normality
to base b to the uniform distribution of {abn} is established), and Chapter 8 (where
the question of ax ...an being a prime for infinitely many n is discussed).
2
Diophantine approximation
Dirichlet'stheorem. Khintchine's theorem. The Duffin-Schaeffer results
and conjecture. The Erdos-Vaaler theorem. The zero-one laws ofCassels
and Gallagher. Cassels' Yrseauences- A crucial lemma (Lemma 2.3).
Overlap estimates. Reduction to GCD sums. Lacunary sequences.
2.1 STATEMENT OF RESULTS
The simplest theorem to prove in Diophantine approximation is as follows.
Theorem 2.1 (Dirichlet) Let a be any given real number, and N a positive integer.
Then
minll/iaKCW+l)"1. (2.1.1)
As we remarked in the introduction, this is best possible, as can be seen by taking
a = (A/r+l)"1. It is best possible in another sense, as shown in Therom 2.3 below.
To prove (2.1.1), consider the distribution of the fractional parts {/za} for 1 ^n^N
in the intervals [0, «5), [«5,25),..., [1 -<5,1) (here <5 = (N+1)"l). If {««} belongs to
the first or last of these intervals, then ||«a|| ^<5, as desired. We suppose that this
does not happen and we shall obtain a contradiction. To do this, note that there
would be TV fractional parts inN—l intervals. Hence by the pigeon-hole principle
there would be two integers /, m, with | {/a} — {ma} | < <5. Thus, if we write n = \l — m\,
we would have {«a}e[0, <5) or [1 — <5,1) and \^n<N. This contradiction establishes
the result. □
It follows from Theorem 2.1 that if a is an irrational number, there are
infinitely many fractions plq in lowest terms with
\<x-plq\<q-2. (2.1.2)
This also follows from the theory of continued fractions, which gives a stronger
result with q~2 replaced by \q~2. Hurwitz [141] improved this further to
5~1/2q~2. This is best possible, as can be seen by considering a = ( —1 + 51/2)/2.
Statement of results
We have a2 + a —1 =0, so, for any fraction m/n:
(m/ri)2 + m/n — \
25
\mln — a| =
m/n + (3
(with£ = (l + 51/2)/2)
1 1
n2 m/n + p'
(2.1.3)
Now, if |m/«-a|<«~2, then, by (2.1.3) (note that m/n + P = 5ll2 + (m/n-a)),
we have
\mln — a\^
1
>z2(51/2 + n-2)
Hence the factor 51/2 is best possible.
It was shown earlier this century by Borel [45] and Bernstein [36] that almost
all real numbers have unbounded partial quotients (an in standard notation) in
their continued-fraction expansion. Writing p„/q„ for the «th convergent to a, we
have, for irrational a,
<
1
QnQn
+ 1
Thus, for almost all real a, there are infinitely many fractions plq with
\cc-p/q\=o(q~2).
Put another way,
liminf N min || no. || = 0 a.e.
Khintchine used the results of Borel and Bernstein in conjunction with an estimate
for the growth of qn to prove the following very precise result. We shall present
an alternative proof independent of the theory of continued fractions.
Theorem 2.2 (Khintchine) Let ij/(x) be a positive continuous function and suppose
that x\j/(x) is non-increasing. Then the inequality
m
a —
n
<A(")
n
(2.1.4)
has infinitely many solutions in integers m,n>0 for almost all real numbers a if
I m (2.1.5)
n=l
diverges. However, if the sum (2.1.5) converges, then (2.1.4) has only finitely many
solutions for almost all real a.
26 Diophantine approximation
Clearly, if (2.1.4) has infinitely many solutions, there will be infinitely many
fractions mln in lowest terms satisfying (2.1.4), since x\j/(x) is non-increasing.
It follows that almost all real numbers have infinitely many approximations of
the form
. |a-m/«|<(«2log«)_1, (2.1.6)
but only finitely many of the form
|a—m/«|<«~2(log«)~1-€
for any € > 0. Of course, this says nothing about approximating to a uniformly as
in Dirichlet's theorem. In fact we have the following result.
Theorem 2.3 For almost all a, Dirichlet's theorem is best possible in the sense that
limsupiVmin ||a«|| =1. (2.1.7)
It turns out that the 'almost all' of Theorem 2.3 includes all those a for which
(2.1.6) has infinitely many approximations.
Theorem 2.2 is included in Theorem 2.5, Corollary 1 below, but first we prove
a very simple result.
Theorem 2.4 Let \j/(ri) be a non-negative-valued function such that
I m^ (2.i.8)
converges. Then there are only a finite number of solutions to the inequality
m
a
n
<-^, (m,«) = l, n^\. (2.1.9)
n
for almost all a.
Proof Clearly, as with all the problems in this chapter, we may restrict our
attention to ae [0,1). If a has an approximation of the form given by (2.1.9), then
so does a + r for any integer r with m replaced by m + nr. Of course, (m,n) = l
implies that (ra + w,r) = l. We write <f„=[0, l)n«^, where
n_1 fm-\j/(m) m+\j/(m)s
(m, n)= 1
If (2.1.9) has infinitely many solutions for some a, then a belongs to infinitely many
of the £„. However,
00 °° (o(ri)
X Wn)<2 X .K*)— (2.1.10)
n=l n=l n
Statement of results 27
by a simple calculation. By the hypothesis of this theorem the right-hand side of
(2.1.10) converges, and so, by Lemma 1.2, almost all a belong to only finitely many
of the in. Thus the proof is completed. □
In higher dimensions Khintchine extended his theorem to simultaneous
approximation, and clearly Theorem 2.4 can be extended almost immediately. We
shall obtain a stronger result than Khintchine's from Theorem 3.8 below (see also
Theorem 4.6).
Duffin and Schaeffer in 1941 [77] made the following conjecture, which has
provoked much research and remains to date one of the most important unsolved
problems in metric number theory.
Conjecture (Duffin and Schaeffer) Let \j/(n) be a non-negative-valued function such
that the sum (2.1.8) diverges. Then (2.1.9) has infinitely many solutions for almost
all a.
We shall give a slightly different formulation of this conjecture later and prove
it for certain special cases. We shall also prove the corresponding conjecture in
higher dimensions. The theorem Duffin and Schaeffer themselves proved is as
follows.
Theorem 2.5 The Duffin and Schaeffer conjecture is true for \j/(n) when, in
addition to the divergence of (2.1.8), we have
limsup( X ij,(n)—)( £ ^(n)) >0. (2.1.11)
N-oc \n=l n /\n=l /
Corollary 1 Let \J/(n) be a positive-valued function such that ny\j/(ri) is non-
increasing for some constant y, and suppose that the sum (2.1.8) diverges. Then there
are infinitely many solutions to (2.1.9) for almost all a.
Corollary 2 Let ij/(p) be any non-negative function defined on the set of primes
such that
p prime
diverges. Then, for almost all a, there are infinitely many solutions to
|a — mlp\ < i{/(p)/p, p prime.
We shall deduce Corollary 1 in Section 2.3. Corollary 2 follows immediately
from the theorem, since cp(p)/p = \—p~1 for primesp, and hence
limsupf X iKp)—)( I HP)) 1=1-
N-oo \p<N P J\p^N J
We shall give some further corollaries to Theorem 2.5 later.
Erdos [92] proved the Duffin and Schaeffer conjecture to be true when, for
some € >0, \j/{ri) takes on only the values 0 or €/«. Vaaler [265] modified the Erdos
28 Diophantine approximation
method to obtain the following result. We will present the Erdos-Vaaler argument
with some technical simplifications.
Theorem 2.6 The Duff in and Schaeffer conjecture is true when il/(ri) = 0(\/ri).
A slight strengthening of this result, which we shall not reproduce here, has
been given by Vilchinskii [267]. We shall make use of the following result to prove
several theorems.
Theorem 2.7 (A) Let \j/(n) be a sequence of non-negative reals. Then the inequality
\an-m\<il/(n) (2.1.12)
has infinitely many solutions for either almost all a or almost no cc.
(B) The conclusion of part (A) holds with the additional condition (m, n) = 1
imposed in (2.1.12).
Of course, it is Theorem 2.7(B) that is required for the above results. The proof
of Theorem 2.7(A) is simpler, however, and should provide the reader with some
insight into the proof of Theorem 2.7(B). Part (A) is due to Cassels [53], part (B)
to Gallagher [104]. The theorem shows that zero-one laws operate in both the
problem of approximation by all fractions and the problem of approximation by
reduced fractions. We indicated in Chapter 1 why this might be expected. The
result of Lemma 1.6 can be used to establish Theorem 2.5 with another condition
imposed on ij/in). To avoid this restriction, a more subtle approach is required,
but one which still uses the basic idea inherent in Lemma 1.6.
At first sight, one might imagine that the divergence of
oo
I m (2.1.B)
n=l
might guarantee the existence of infinitely many solutions to (2.1.12) for almost
all a. This is shown to be false by the following result of Duffin and Schaeffer.
Theorem 2.8 There is a non-negative function ^/{n) defined on the positive integers
for which (2.1.13) diverges, yet
\aL-mln\<\l/(n)ln (2.1.14)
has only finitely many solutions for almost all cc.
Catlin [60] has made the following conjecture in this area.
Conjecture (Catlin) // \j/(ri) is a non-negative function defined on the positive
integers, then the divergence of
* cp{n) \lf{mri) ......
> max (2.1.15)
„=i n m>l m
is the necessary and sufficient condition for (2.1.14) to have infinitely many solutions.
Zero-one laws 29
It is not difficult to establish that the divergence of (2.1.15) is a necessary
condition for (2.1.14) to have infinitely many solutions. Also, it is easy to show
that the conjecture is true if the Duffin and Schaeffer conjecture is true. Catlin
[60] claimed that his conjecture is equivalent to the Duffin and Schaeffer
conjecture, but his proof that his conjecture implies the other contains a serious
flaw, as first pointed out by Vaaler (page 530 of [265]). The present author has
not seen a correct proof of his claimed result.
Finally, we mention a result given by Cassels [53]. Given an infinite sequence of
positive integers an, write Mr for the number of fractions of the form j/ar (0 <j<ar)
which are not of the form klas, with s < r. Cassels calls an a ^-sequence if
1 M
liminf- £ -^>0. (2.1.16)
He proves that if \j/(ri) is non-increasing, an is a ^-sequence, and (2.1.5) diverges,
then there are infinitely many solutions to
0^ccan—m<\l/(n), m,«eZ, (2.1.17)
for almost all a (we remark that any of the previous results of this chapter could
be restated in the 'one-sided' form of (2.1.17) with only slight alterations to the
proof). We shall prove Cassels' result in Section 2.3, since it required only a minor
modification to the proof of the Duffin and Schaeffer theorem.
Since the sequence an = bn is a ^-sequence for any integer b^2, there are
infinitely many solutions to
0^{6na}<—!—.
nlogn
Combining this result with the obvious convergence case, we thus obtain the
following result on the occurrence of long blocks of zeros in decimal expansions.
Let cc = A-ala2---in scale b. Then, for almost all cc, there are infinitely many n
such that an=an+1=--=an+m = 0, for
Plog*J
when P = \. On the other hand, there are only finitely many such n when f}> 1.
2.2 ZERO-ONE LAWS
We shall follow Gallagher's method here, which in the author's opinion is a neat
modification of Cassels' original argument (Lemma 9 in [53]).
Lemma 2.1 Let Jk be a sequence of intervals and Sk a sequence of measurable sets
such that, for some 5 € (0,1),
m =
Sk^Jk, l{£k)>5Wk\ A(J^0.
(2.2.1)
30 Diophantine approximation
Then the set of points which belong to infinitely many of the Jk has the same measure
as the set of points which belong to infinitely many of the Sk.
Proof Clearly we can discard intervals with length zero and alter the outcome
only by a set of measure zero (indeed a countable set!). We may thus suppose that
^(J^O for all A:. Write
>= n 0 a, ^n= u ^
n=lfc=n k=N
The set on which cLeJk has infinitely many solutions but a € £ k has only finitely
many solutions is
N=l
We therefore wish to prove that M$N) = 0 for all N. The proof is by contradiction.
Suppose that AC#w)>0. Then, by Lebesgue's density theorem there is an xe^N
(a point of metric density) such that, for any sequence of intervals #n with *€</„,
^(A) >0, and A(/„)^0, we have
limiW=1
Since xeJn for infinitely many n, we may take the £n as a subsequence of the
Jni and so deduce that there are arbitrarily large values of n with
Wn^N) >(1 - S/2)X(Jn). (2.2.2)
Now, for n^N, £nn<$N = 0. Since Sn^Jn, this gives
KSn)>K*n) + WNC\fN)>5KJH) + WeNnfH) by (2.2.1)
Ml + S/2)X(JH) by(2.2.2).
This is our desired contradiction, since X(Jn)>0, and so we conclude that
H$N) = 0 for all N as desired. □
Remark Cassels [53] omits the necessary hypothesis A(./n)->0 in his statement
of the result. We shall apply the result in the following form (Lemma 9 of [53]).
Corollary Let fin be a sequence of reals, and \j/(n) a non-negative function with
if/(ri)-+0 as «->oo. Write &fl = {xe U: \x—Pn\<il/(ri)n has infinitely many solutions}.
Then ?.(@lt) = l(@l) for every fi>0.
This follows immediately from Lemma 2.1, since we can suppose, after a
change of variables, that 0</z<l. We take
Zero-one laws 31
We use the following notation from ergodic theory for convenience (compare
our discussion at the end of Section 1.2).
Definition A transformation T: [0,1) -* [0,1) is said to be ergodic if for every
measurable set with £T=£, A(<f) = 0 or 1.
Lemma 2.2 Let q and s be integers with q^2. Then the transformation Tgiven by
xT={qx+s/q}
is ergodic
Proof Let S be a measurable subset of [0,1) with ST=S. If /(<f) = 0, there is
nothing to prove, so we assume l(£)>0. Hence there is a point of metric density
a#0, aeS. Consider the set
@n = (a-q-n/2,a + q-n/2)n£.
For all large n we have ^„cz [0,1). Since a is a point of metric density, if
we have 0„->l as «->oo. Now, if ST=S, then STn = S with
xTn = {qnx+s/q}.
Because 01 n is contained in an interval of length q~n, we have X(qn0ln +
s/q) = qnl(&„), and qnMn+s/q is contained in an interval of unit length. Thus
X{S) = l{STn)^?,{mnTn) = 0n^\ as «-*oo.
Hence £ has measure 1, which completes the proof. □
We are now in a position to prove Theorem 2.7(A). Clearly we can restrict our
attention to [0,1). Suppose first that \l/(n)ln does not tend to zero as n tends to
infinity. Then for certain integers nl,n2i... we must have i/finj) ^ 1. Hence (2.1.12)
has infinitely many solutions for every a.
We can now suppose that \l/(n)/n^0 as «->oo. Write
E(t) = {ae[0,\):\a-m/n\<2til/(n)n-1 i.o.}, f = 0,l,...
By the Corollary to Lemma 2.1, we have A(E(0)) = A,(E(t)) for every t^\. Put
f = 0
Then a € S <=> {2a} e S. Hence, by Lemma 2.2, X(£) = 0 or 1. If A(<f) = 0, then
X(E(0)) = 0. If X(£) = 1, then, since E(s) c E(t) for s < t,
lim ;/£(/)) = 1.
32 Diophantine approximation
Because X(E(t)) is independent of /, we conclude that l(E(t)) = 1 for all t, and so
A(2s(0)) = l. This completes the proof. □
Clearly there is a problem in using the above argument when the condition
(m,ri) = \ is imposed. We made use of the fact that if |a—m/n\<rj, then
12a—2mln\ <2rj and 2mln was an admissible fraction. Once we require numerator
and denominator to be coprime, we are in difficulty with even n. Of course, 2
could have been any integer t> 1, so that the difficulty would then arise for those
n with (n,t)>\. We overcome this problem by considering all primes p, and,
depending on the power of p dividing «, using the three transformations:
aT=pcc, ccT=pcc + -, ccT=a+-.
P P
We note that the first two of these transformations are ergodic.
Tacitly in the sequel we shall make use of the fact that if
\nfi — m\<r\ with (ra,«) = 1,
then there is an integer m' such that
\n{fi} —m'\<rj and (m',n) = 1.
For Theorem 2.7(A) we can again suppose that \j/(ri)ln-+0, since
\tiol — m\ < dn d\n \na — m\<dn
(m, n) = 1 d\m
= YJti(d){2n5d-l + 0{\))
d\n
= 2S(p(n) + 0{rf) for any € > 0.
Since (p(n)/n»n~€ (see Hardy and Wright [111], Theorem 327), if \J/(n)/n-H>0 the
inequality (2.1.12) has infinitely many solutions for every a even with (m,«) = l.
For each prime p and positive integer r, put
j^(/>r) = {ae[0,l): \noc-m\<pr~^(n), (m,n) = \, i.o., p)(n\
@(pr) = {(xe[0,1): \n(x-m\<pr~1il/(n), (m,«) = l, i.o., p\n,p2)(n},
#(/?r) = {ae[0,l): \na-m\<pr~ V(«), (m,n) = l, i.o., p2\n}.
Clearly, if we write
<f = {ae[0,1): \na—m\<\j/(n), (m,ri) = \, i.o.},
then
g = s/(p)ua(p)u<g(p) (2.2.3)
Zero-one laws 33
for every prime p. We shall demonstrate that for every prime p the sets s#(p) and
&(p) have measure 0 or 1. It will then remain to discuss the sets #(/>)•
The sets s/(pr), &(pr) behave, for fixed p, like E(t) in the proof of part A (with
r in the role of /). We have
j/(pr)^j/(pr+l), @(pr)^@(pr+l). (2.2.4)
We put
r=l r=l
By the Corollary to Lemma 2.1, we have X(<srf(pr)) = X(jrf(pr+1)). Thus, in view of
(2.2.4),
Similarly, l(c§(p)) = l(g8(p)). Now define T by xT ={px}, so T is ergodic by
Lemma 2.2. If ae<stf(pr), then there are infinitely many pairs of integers m, n with
| pan —pm\ <prij/(n) and (m, /i) = 1, /7/fw
The two conditions {m,n) = \,p)(ncombine to give imp,n) = \, pjfn,so that
{/>a}e^(//+1).
Hence #"(/?) is invariant under T, and so has measure 0 or 1.
We establish the same property for &(p) by considering the transformation
olT={pol+ \lp).
Evidently, if
\na — m\<pr~ 1\J/(n), p\n, p2J(n, (m,n) = \,
then
\n(ap+ \lp) — (pm + n/p)\<pr\l/(ri), p\n, p2J(n, (pm + n/p,ri) = \.
(Note that it is vital here that p2Jfn.) Thus <g(p)T=<g(p), so that a(^(/?)) = 0 or
1 by the ergodicity of T.
We have thus shown that, for every/?, l(s#(p)) = 0 or 1 and 2(^(/?)) = 0 or 1.
Thus, if ),(£)$ {0,1}, then k(&) = M${p)) for every prime p. Now the set <#(p) is
periodic with period \lp. To see this, note that if
| net — m\ < \J/(n) with (m, n) = 1,
then
\n(a + \lp) — (m±nlp)\<\j/(n) with (m±n/p,n) = l.
(Note that it is vital here that/?2|«.) Thus, for any interval Jp of length 1//?, we have
K£r\Jp) = KJP)K$) • (2.2.5)
34 Diophantine approximation
Now, if X($) > 0 we can find a point of metric density x e £. Let Jv be an interval
of length \lp centered at x for every prime p. Then, by the definition of a point
of metric density,
lim^f)=1. (2.2.6)
p->co ^ p
Comparing (2.2.5) and (2.2.6), we discover that k(£) = \i which completes the
proof of Theorem 2.7(B). □
Before proceeding further, we pause to consider a possible approach to solving
the problems in this chapter and see that it fails. This may also help us to consider
what properties a sequence must have to be a counterexample for the Duffin and
Schaeffer conjecture. Let \jt(n) e [0, |] and put
n = i n £x n
(if we do not require (m,«) = l, then omit (p(n)/n, r(n) in the above). We assume
that N(oo) diverges. We note that M(R)/N(R)-+0 as R-+oo. Now let A<B and
write j&(V, R) for the set of a € [A, B] for which N(a, R) ^ V, where N(oc, R) counts
the number of solutions to
| an—m\<\j/{n), (m,n) = \, n<R.
Clearly, for any pe[A,B], any €>0,
N(x,R)da = 2eN(R) + 0(M(R)),
where the implied constant depends at most on A and B. Now suppose that
Pejrf(V,R), and for every ae[A,B] we have N(ol,R)^KN(R), where K is
independent of a e [A, B]. Then
2aN(R) + 0(M(R)) =
CP + e
N(a,R)d<x +
p-e
CP+e
N(a,R)da
P-€
^ KN(R)(2<l - Kst<y, R)n [p - €, fi + e])) + a(s/(V, R))V.
Rearranging (and using the trivial bound ?^(V,R)) ^B — A), this gives
}^(V,R)n[p-eJ + e]) {K-2)N(R) + Q(M(R) + V{B-A))/c)
2€ KN(R)
Since the limit of the right-hand side above is 1 — 21K as R tends to infinity, and
stf(y,R + \)<=d{V,R), we deduce that
,. M^(K)n[jg-6,jg + €])
lim <1,
Zero-one laws
35
where
R=l
Hence ?.(<srf(V)) = 0, for it has no points of metric density. Thus
,K=1
and so for almost all a there are infinitely many solutions to our inequality.
The problem with the above argument is that the inequality N(a,R)^KN(R)
is quite probably false for certain a. To give an extreme example, let a = 21/2
and let qj be the denominator of the convergent to the continued-fraction
expansion of a. Then put
[0 otherwise.
Since qj<tBJ for some B for any a with bounded partial quotients, and indeed we
can take 5 = 3 for a = 21/2, we have (using (?(«)/«»(log log «)-1)
X «A(«) » £ -n—: = °°-
„=! n j=2;logy
However, N(a9 R) = R for all R, whereas N(R)«log R. The argument thus fails for
this sequence, although the desired conclusion may be reached by an alternative
argument (Theorem 2.9 below works, since qi is a lacunary sequence). If we
remove the condition (m,«) = l, we discover that N(a,R) becomes large near
rational points with fairly small denominators, even in the simplest case \j/{n) = \ln.
Even though this argument is not applicable here, we remark that it can be used
in other situations, one of which we shall consider in Chapter 8.
We complete this section with a result which will be used in conjunction with
the zero-one laws. We shall use the following lemma to prove that the set on
which an inequality has infinitely many solutions has positive measure. The
zero-one law will then guarantee infinitely many solutions for almost all x.
Lemma 2.3 Let X be a measure space with measure fx such that fi(X) is finite. Let
Sn be a sequence of measurable subsets of X such that
f>(0 = oo. (2.2.7)
n=l
Then the set E of points belonging to infinitely many sets Sn satisfies
M^^limsupfi^^Yf I rt^.no) '• (2-2.8)
N-oo \n=l / \n,m=l /
36
Diophantine approximation
Remark In one sense this result is best possible, since when
n, wi= 1
^n=l
(as often happens, and always when the Sn are independent), (2.2.8) gives
H(E)^ii(X). On the other hand, it can be the case that the right-hand side of
(2.2.8) is zero, yet ii(E) = ia(X). Ingenuity is then required to remove some parts
of the sets Sn which genuinely have a large overlap with other members of the
sequence (see Chapter 4).
Proof Without loss of generality, we may suppose that pi(X) = \. Write
j = m
M(m,n)= £ fitfj), V(m,n)= £ £ ^n^);
j = m j = m k = m
also let Xj(x) be the characteristic function of &-p and let xm,nix) be the
characteristic function of Enm. We wish to prove that
lim (lim KE"j) > limsup^y, (2.2.9)
for then (2.2.8) will follow. In view of the divergence of (2.2.7), we have, as n -* oo,
Af(/w, «)=M(0, n) + 0(m),
K(m, n) = V(0, n) + 0(mM(O, /i)) + 0(m2).
It therefore suffices to establish that
M(m, ri)2
KEnJ>
V(m, n)
(2.2.10)
to prove (2.2.9), since M^m) is a non-decreasing function of n. The inequality
(2.2.10) is a simple consequence of the Cauchy-Schwarz inequality, as we now
demonstrate.
We have
M(m, n) =
Z Xj(x)dx= | xm.n(x) Z Xj<x)dx.
X j = m
j = m
Thus, by the Cauchy-Schwarz inequality,
(M(m,n))2<|
Xm,n(x)dx
I* n
Z XjMXkMdx
X j,k = m
= fx(Enm)V(m,n).
This establishes (2.2.10) and completes the proof as required.
□
The Duff in and Schaeffer theorem 37
2.3 THE DUFFIN AND SCHAEFFER THEOREM
We first remark that the Duffin and Schaeffer proof appears to contain an
oversight which limits their result to cases where \l/(n)^\ for all n (or, with a little
modification, \j/(ri)^C for some C for all n). The problem that arises will soon
become apparent. Let
^ = * * fm-xl/{n) m + ij/(n)\
m=l \ n n J
(m,n)= 1
Here the individual intervals making up £n are to be interpreted modulo one. Thus,
by way of illustration, ((8-2)/9, (8 + 2)/9) is taken to mean (6/9,l)u[0,1/9).
Clearly, if \J/(ri)^j for all n, then A,(£n) = 2<p(ri)/n\J/(ri). However, if ^(«)>i, then
the intervals making up $n can overlap. The difficulty this introduces was first
overcome in a paper by Pollington and Vaughan [225]. We first remark that we
may assume that <p(ri)ln i}/(ri) < £. To see this, note that replacing \j/(ri) by vs\m{\j/{n),
n{6(p{n))~x) maintains the divergence of
2, —tA(«),
n = i n
and does not increase the size of the set on which our inequality (2.1.9) has
infinitely many solutions. This observation avoids the appeal made in [225] to a
deeper result for the case (p(n)/n\J/(n)>j.
We now show that
1 (p(n) (p(ri) 1
K£n)>~ — Hn) when^>(«K-, (2.3.1)
in no
working in a similar manner to Pollington and Vaughan. Let t = <p(n) and suppose
that al,a2,...,at are the integers coprime to n between 1 and n. A simple
calculation yields
W = \( £ min(2^(«),aj-aj. x) + 2mm(ij/(n), 1)
To obtain this, we note that the distance between centres of consecutive intervals
in Sn is (a-j—a-j-^ln. Writing at+l = n, we find that
K*n)>- I mm(2iKn),aJ+l-aj)
nJ=l
Mn)
^^^-{(p{n)-A{n)), (2.3.2)
n
38 Diophantine approximation
where
Ain)= X 1.
aj+l-aj^>p(n)
Let H=\l/{n). Then
h^H a=\
(a(a + h),n)=l
= 11 tie) I 1
/i^tf(e,/i)=l a=l
e\n a= -/i(mode)
(a,n)= 1
Z x Me) Z m Z i
/i^tf(e,/i)=l (d,e)=l m=l
e|n d|n md=-/i(mode)
= 11 tie) I M<0^
e\n d\n
/i^tf W (e,/i)=l g Pie \ /*>
e|n
*7h * plnV P-l
Pi*
^■n
f(p(n)\2 y (p(n/(h,n)) n
\ n ) h^H nl{h,n) q>(n)
V « / i.<h^) \ n
where the final inequality follows from a well-known result in elementary number
theory (stated explicitly as Lemma 2.5 below). Substituting this bound for A(n)
into (2.3.2) then gives
;W^(«)(i-3—*<4
n \ n J
which establishes (2.3.1).
The basic idea of the proof of the Duffin and Schaeffer theorem is to estimate
X{Snr\S„) in such a way that an application of Lemma 2.3 allows us to conclude
that the measure of the set of points falling in infinitely many of the Sn has positive
The Duff in and Schaeffer theorem 39
measure. An application of Theorem 2.7(B) will then complete the proof. For
obvious reasons the bounds for X(£mr\£n) are called 'overlap estimates', and play
a crucial role in the metric theory of Diophantine approximation.
Now suppose two intervals, one in Sni one in Ssi have non-empty intersection.
Thus
^ s s J \ n n J
for some integers r,m with (r,s) = (m,ri) = l. The coprimeness condition ensures
that the centres r/s, mln of the intervals are distinct. We must therefore have
0<
r m
s n
WW
s n
the last inequality arising from the fact that there is some overlap. Hence we obtain
0<|r«—ms\<ml/(s)+sil/(ri)^2ma.x(ml/(s),sil/(n)). (2.3.4)
Many parts of the remainder of this book will be devoted to obtaining satisfactory
estimates for the number of solutions to 0<\m—ms\^A, or to rn = ms, with
various restrictions placed on the variables. These questions are clearly connected
with multiplicative number theory. The first is linked to solving rn—ms=\, an
equation which is of central importance in much number theory. The second is
clearly connected with the divisor function, or some restricted divisor function,
and how these functions behave on average.
We now obtain the simplest bound for the number of solutions to
0<\m-ms\^A, l^r^s, l^m^n (2.3.5)
with n and s fixed. Put n' = n/(n,s), s'=s/(n,s). Clearly there are no solutions if
A<(n,s). Otherwise, we have
rn' = a(mods')
for some a, with 1 <|fl|^/l/(a,.s). There is only one solution in r (mods') for each
value of a, and so sis' values of r for each value of a with 1 ^r^s. Hence (2.3.5)
has ^(2A/(n,s))(n,s) = 2A solutions. It should be noted that we have used the
coprimeness condition only to exclude the possibility that rn = ms. We shall
establish later that the number of solutions including the coprimeness condition
is often «A(p(n)ln(p(s)ls.
Now the intersection of two intervals is no more than the length of the shortest
of the two intervals. Hence, for the intervals under consideration this becomes
2min(\l/(ri)/n, \j/(s)/s). Assembling all our results so far thus yields
(Mn) Msf
K$s n Sn) < 8 max(#(.y), $ W)mm ("^»—
= 8 WW-
40 Diophantine approximation
Thus, writing E for the subset of [0,1) belonging to infinitely many of the Sr, an
application of Lemma 2.3 produces
Hence, by (2.1.11) and Theorem 2.7(B), we obtain n(E) = \. This completes the
proof of Theorem 2.5. □
To deduce Corollary 1 we require the following well-known result, which may
be obtained by partial summation from Theorem 330 in [111] (or easily established
directly).
Lemma 2.4 For any N^\ we have
^<m=t_N+0(logAO (236)
It follows from Lemma 2.4 with partial summation that, for any y<0,
Thus, for all large N,
i^lA+^-nos*)
n=ln ^n=l
Now, if \J/(ri)np is non-increasing for fi = y, then it is non-increasing for all /?<y.
Hence we may suppose that y<0 in Corollary 1 to Theorem 2.5. We have
X —\l/(n)= Z(n^(n))-r+7
n=l n n=l n
N n a>(m)
= X MnW-iKn + lKn + m £ ^
n=l m=im
N o(m)
+ ^(iV+l)(iV+ir X^TT
n=lrn
^I*K")+0(1)
^n=l
by (2.3.7). Consequently (2.1.11) holds and this completes the proof of the
Corollary. □
We may deduce some further corollaries using the following elementary
lemma.
The Duffin and Schaeffer theorem 41
Lemma 2.5 For any N^2 we have
" n 315C(3)JV
Moreover, the error term here is negative.
Proof We note that
+ 0(\ogN). (2.3.8)
Hence
<?(») \i\ p) \l\ p-v h<p(d)
N „ ..2
fi\d) „ , Ar » fi\d) » ix\d)
N u2(d)
=N I J^+0(\) + 0(\ogN)
d% (p(d)d
(note the implied constant here is negative)
="n(1+^)+0(,og;v)
(again the additional implied constant is negative)
ArC(2)C(3) , „n
=N——- + 0(\ogN),
C(o)
which gives (2.3.8). We note that
31503)
2tt4
1.94. □
Definitions Given a set of distinct positive integers si, we define the lower
asymptotic density of si (l.a.d. of si) to be
1 N
liminf- £ 1.
N-oo iV n=l
Similarly, the w/?/?er asymptotic density of j?/ (u.a.d. of si) is
1 "
hmsup- X 1.
N-oo iV n=l
ne.c/
If the u.a.d. of si = c = \.a..d. of j?/, then we say that si has density c. We then
have the following corollary to the Duffin and Schaeffer theorem.
Corollary 3 Let \j/(ri) be a positive-valued function defined and non-increasing on
a set si with positive l.a.d. Then there are infinitely many solutions to
\an—m\<if/(n)i (m,n) = \, nesi,
42 Diophantine approximation
if and only if
I Hn) (2.3.9)
ne.W
diverges.
Proof The 'only if part is a consequence of Theorem 2.4. Now suppose that the
l.a.d. of stf is c and let @ be the set of positive integers for which (p(ri)/n^c/4.
Then, by (2.3.8), for all N we have
n 4
Thus
so that
I 1 ^Nc,
n=l
N Nc
Z i>Y
n=l J
for all large N.
Now put
.^(n) if nes#r\@
'0 otherwise.
Then
£*V 4.
*'(«)^7 !*'(*)
=jI(!Kb)-!Kh + 1)) Z 1+^(^+1) Z 1
^"n=l m=l ^ m=l
wi e .t/ndf m e .v/r\26
(extending the definition of \j/ if necessary to be non-increasing for all positive
integers)
^ I Wn + \)-*{n))n+^jt{N+\)N+0<\)
c2 N
= TX(1+0(D)I <A(«)
^ n=l
^^(i+o(D) z nn).
1Z n=l
Hence (2.1.11) holds for \j/' and Corollary 3 follows. □
The Duff in and Schaeffer theorem 43
There are many sequences to which Corollary 3 may be applied. For example,
the hypothesis holds for si being an arithmetic progression or the set of
square-free integers. These results can be proved more directly, and follow from
the Duffin and Schaeffer theorem. Taking i}/(ri) = \/(2n) gives the result that given
any arithmetic progression a(modq) with (a,q) = l, almost all a have infinitely
many convergents to their continued-fraction expansion with denominators in si
(this seems to have been first stated by Hartman and Szusz [130]). The same
conclusion holds if si is the set of square-free integers or, by Corollary 2, primes,
or even primes in arithmetic progression. Later (Chapter 6) we shall consider what
happens when numerator and denominator are both restricted to given sets.
Further results can be obtained as corollaries to the Duffin and Schaeffer
theorem, for example with si being the set of numbers properly represented as
the sum of two squares (a set with density zero, of course). One can take si to
be the set of values taken by a polynomial whose leading coefficient is positive.
This follows from the following lemma, whose proof we leave to the reader.
Lemma 2.6 Let P(x) be a polynomial with integer coefficients, whose leading
coefficient is positive. Then, for every € > 0,
£ SM-Wi-^Wat), (2.3.10)
n=l n p>2\ P J
P(n) > 0
where r{p) is the number of solutions to the congruence P(n) = 0 (mod p), 0^«</?.
The product in (2.3.10) is convergent, since r(p) is bounded by a constant
depending only on P(x). In all the examples we have considered so far (with the
exception of Corollary 2), xj/in) is non-increasing on a sequence which is growing
at a steady rate. When r is an integer not less than 2 and
sf = {rm: m = 0,l,2,...},
no restriction is required for \J/ save the divergence of (2.3.9). This follows since
here,
limsupfi^^Yi^r—>o.
JV-oo \n=l n /\rt=l / r
ne.<V neV
In Sections 2.4 and 2.6 we shall develop different methods in order to attack the
Duffin and Schaeffer conjecture itself.
We end this section with the proof of Cassels' result. We must now replace
Sm with
a°. -1 (m m + \lt(ri)\
m/a„^k/as for s<n
44 Diophantine approximation
We also write
1 N M
C=^liminf X—.
2 N-oo n=l an
By (2.1.16) we have C>0. Also,
n=l n=l
for sufficiently large N. On the other hand, we recall that we used the coprimeness
of m and n in the definition of Sn only to exclude the possibility of coincident
centres of intervals in our treatment of Snr\Ss. Clearly this possibility is excluded
here by the condition mlan^klas. We therefore obtain X{SsnSn)^^/{s)^/{n) as
above. The proof is then completed by Theorem 2.7(A) (which can be made
'one-sided' with no difficulty) and Lemma 2.3.
2.4 VAALER'S THEOREM
We begin by improving the crude bound we previously obtained for the overlap
estimates. The estimate we need (Lemma 2.8) was first stated by Strauch [225],
although essentially the same bounds were given earlier by Erdos and Vaaler.
A different proof from the one that we present has been given by Pollington and
Vaughan [225]. We first require a simple bound from sieve theory. All contants
implied by the « notation in this section are absolute.
Lemma 2.7 Let X be a positive real exceeding 1 and n a non-zero integer. Then
i >«*n(i-;}
n^X p\n \ PJ
(m,n)=l p^X
Proof See Theorem 2.2 of [108].
Lemma 2.8 For m^nwe have
X(SmnSn)« Ufim)MH)P(m,n), (2.4.1)
where
(2.4.2)
\(m,n)2
p>D(m,n)
and
max(miKH),#(m)) ,~ A ~.
D(m, n) = . (2.4.3)
(m,n)
Vaaler's theorem 45
Proof Write
A = m2ix(m\l/(n),n\l/(m)), t = (m,ri), a = A/t, f=m/t, g=nlt.
We wish to bound the number of solutions to
\rm— sn\<A with (r,n) = (s,m) = l, r<n, s<m, (2.4.4)
which becomes
\rf— sg\<a with (r,n) = (s,m) = \, r<n, s<m. (2.4.5)
Say, for b>0,
rf-sg = b. (2.4.6)
Let x denote the inverse of x (modg), that is, xx = \ (mod g), l^x^g — 1. Then
from (2.4.6) we obtain
m
r=bf+dg with OsS</+| -f- |<?.
Let
n ff~X
Since we must have (b,fg) = 1 = (rs, t) to satisfy the conditions of (2.4.5), we find
that the number of solutions to (2.4.6) is
i**) ii =<n(i-^)
e\t d p\t \ F J
(bf + dg)(bq + df)sO (mod e)
(where
ri i(p\bfg
[2 otherwise
«'nH) nB)_1n H
P\t \ P/p\(t,b)\ PJ p\(t,fg)\ Pj
pXfg
Hence the number of solutions to (2.4.5) is
«<nH)nH) i n(.-T
p|t \ P/p\(Ufg)\ PJ 5=1 p\(t,b)\ Py
p>3 (b,fg)=l
P*f9
46 Diophantine approximation
Now the sum over b above equals
y J*V) y u<yAd)a i-r A !
y <p(<0 ^ <K j£ <p(d) d }}g V p,
(b,fg) = l p^a/d
b = 0(modd)
using Lemma 2.7,
/jV)alog(</ + l) / 1
* *«**> pi/A ^
p^a
«
«z^wn(i-;)«n(i-J
p<a p^a
Hence the number of solutions to (2.4.5) is
-n(.>(.-;)n(.-;)
p<a\ PJ p\t \ PJ p\(t,fg)\ PJ
p\fg ptfg
^H)^B)yH)n(4
p\fg pJrfg p\fg
PXfgn p\fg
= P(m, n)
n m
We can now deduce (2.4.1) using the method of the previous section. □
Clearly the problem we need to overcome is the possible influence of P(m, ri)
on the estimate. Here we shall use a method introduced by Erdos, whereas a
different approach will be tried in Section 2.6. Following Erdos, we write g(s) for
the least positive integer such that
pT* P 2
P>9(s)
Vaaler's theorem 47
Two facts we need that follow immediately from this definition are
4t «log(2g(5)), and [1 f1"") ' «*• <2-4-7)
p>g(s)
We thus obtain the following result.
Corollary to Lemma 2.8 Write
t = t(m, n) = max(g(ra/(ra, n)), g(n/(m, n))).
Then
its r,/> \ ~ jWJWn)\og(2t) when \<D(m,n)<t
KSmnSn) « |WJA(0 when t<D^ny (2A8)
We now explore further properties of g(s) in the next two lemmas.
Lemma 2.9 Let X^\ and t a positive integer be given. Then
g(n)>t
Proof We follow Erdos [92]. A sharper and more general result is given by
Vaaler [265]. We first consider those integers not exceeding X which have at least
2/ distinct prime factors in the interval [t,E), where £" = exp(/2). The number of
such integers is no more than
X / y V\2t X(3 log(2Q)2t A]
Now let Pi<p2<-<P2t be the first 2/ primes exceeding /. By the prime
number theorem,
| lo^logr
;=i log?
Hence, for all large t, the left-hand side of (2.4.9) is smaller than J. Thus, for an
integer n < X with g(n) ^ /, but not having 2/ distinct prime factors in the interval
[t, E), we obtain
pTnP ^
48 Diophantine approximation
Therefore
n^X L' n^X p\n P
g(n)>t g(n)>tp>E
x A
« —+4
p>E
x Av _ 1 x
p>E
which completes the proof. D
Lemma 2.10 For positive integers t, q we have
X -j«log2f.
d\q d
9(q/d)**t
Proof We use (2.4.7) to note that
qld
Hence
log2f»l when g(qld) <f.
v 1 „ 1 (p(q/d)
giq/d)^t
= ^5>(r) = log2f.
a A*
This proves Lemma 2.10 (the argument we have used here is much simpler than
those used in [92] and [265]). □
Proof of Theorem 2.6 Our aim is to prove that, for all N,
and so apply Lemma 2.3. In view of the Corollary to Lemma 2.8, it suffices to
demonstrate that
* MmMn)<p(nMm)lQZ2t ^ / » MO^X (2 4 ,0)
Vaaler's theorem 49
where * represents the conditions
m<n^N, - <D(m,n) <t(m,n). (2.4.11)
There are four cases to consider. We write r = (m,ri) and s=mlr in the following.
Case (i): t=g(m/r), D{m,n)=m\j/{n)lr. The sum we are dealing with is thus
< X ^Hn) X X *(r5)(log2/)
n=l n r\n 1/2 <s\l/(n)<t
9(n/r)^t g(s) = t
« £—*(*> Z I (logaXr*)-1
n=l n r\n l/2<s\l/(n)<t
g(n/r)^t g(s) = t
(using \j/(m) = 0(m~1))
» cp(n) (log2Q2
« > w(«) > -— by lemma 2.9
9(n/r)<f
«—s-— \ Mri) by Lemma 2.10.
Since
y(l0g2/)3
?!
converges, this is a satisfactory estimate.
Case (ii): t=g(nlr), D{m,n)=n^/{m)lr. The proof for this case is analogous to
case (i), with the roles of m and n reversed.
Case (iii): t=g(mlr), D(m,ri)=n\j/(m)/r. For every integer h we put
x¥{h) = {m: 2h^mil/(m)<2h+1}.
Since i//(q) = 0(q~1), we have
Z 2*«1.
/i
We commence the proof for this case by fixing / and h and restricting our attention
to me*P(/z). We then have
£—*(*) Z Z *(r5)(log20
n=l n r\n 1/2 <^(rs)n/r<t
g(n/r)^t rse^/i)
50 Diophantine approximation
«2*£—«») I I -Oog 20
n=l H r\n (2hn/tr2)<s<(2h+2n/r2)rS
9(n/r)^t
2ft(log2Q3 £ p(n)
by Lemmas 2.9 and 2.10. Again, this is a suitable estimate after summing over h
and t.
Case (iv): t=g(n/r), D(m,n)=m\l/(n)/r. This case may be considered as case (iii),
with the roles of m and n interchanged.
Combining our estimates, we obtain
m<n^N \n=l n J n=l n
which with Theorem 2.7(B) and Lemma 2.6 completes the proof. □
It is worth while to consider the various points where we used \j/(q) = 0(q~1)
and see how they contributed to the proof. Roughly speaking, we note that the
range of values m, n for which the log 2t factor appears is small and, whichever is
taken as the inner range of summation, nothing is lost by taking i}/(q) = BIq for
every value in this range. We then rely on the fact that P(m, n) is, on average, no
more than a constant. Of course, a vital role is played by \j/{rs) factoring out as
«(l/r)(l/s) to make the proof work, especially since there is a complicated
interdependence of the summation variables brought about by the greatest-
common-divisor factor, which necessitates the sum
which appears above. The reader should have no trouble in verifying that the
condition \j/(q) = 0(q~1) may be relaxed to
q>(qMq) = o(t ^^nn-
Finally we remark that Vaaler deduced his theorem by a simpler argument
(plus an appeal to Gallagher's zero-one law) from the following result.
Write
Proof of Theorems 2.3 and 2.8
51
where 2t is any finite subset of the integers exceeding 1. // \j/(ri) < Cn~1 for all n
and some C>0, then there exists a real number rj0 such that the following condition
holds: ifO< A( Jf) ^ rj0, then
X £ A(<fmn<f„) « A(^)2(loglog(A(^)-1)):
ne & me &
2.5 PROOF OF THEOREMS 2.3 AND 2.8
We start with Theorem 2.3. By Theorem 2.2, for almost all a there are infinitely
many fractions alq with (a, q) = 1 and
a
a
o
<{q2\o%q)
-l
Let rls be the convergent to the continued fraction expansion of a with largest
denominator less than q. Since alq is also a convergent (at least when q > e2, see
Theorem 184 in [111]), we have
a
and so
Hence
r
a —
s
1
+
a
a
qs
\sol — r\ = s\
a
a —
However, for n ^q — 1, we have \\na\\ ^ ||sa||, and so
(q-\) min ||«a|| >—I 1-
1
n<g-i Q \ log?,
Since (2.5.1) holds for infinitely many q, the result follows.
(2.5.1)
□
The reader will have no difficulty in proving that Dirichlet's theorem can be
improved for a real number a if and only if a has bounded partial quotients in its
continued-fraction expansion. We note that the proof of Theorem 2.3 cannot be
generalized to simultaneous approximation, or to linear forms in two or more
variables, since there are no analogues of continued fractions in these situations.
However, Davenport and Schmidt [71] have proved the corresponding results, but
with much more work than is needed for the case discussed above.
Before we prove Theorem 2.8, we required the following lemma. Our proof
closely follows Duffin and Schaeffer's original argument.
52 Diophantine approximation
Lemma 2.11 Let i? ^ 1 > € > 0 be given. Then there is a sequence aq of non-
negative numbers such that
q
ag = 0 for q ^ R and for q > N(R, €),
but the inequality
a
a
< ^ (2.5.2)
can be satisfied only for a set of cue [0,1) of measure €.
Proof The basic idea is to make many of the fractions coincide, so we will take
aq non-zero only for q dividing some large number N. To do this, let rj = e/2 and
let pl9... ,pk be the first k primes exceeding R such that
n(l+-)>l + ^_1. (2.5.3)
Now write
and let
If we put
N= ft Pi.
_jqrj/N if n>\, n\N
q [0 otherwise.
then X{S^ = 2aai while
q q J \q N q NJ
so that Sqc £N. It follows that (2.5.2) has solutions only when aeSN, and so for
a in a set of measure €. On the other hand,
l«,4l»4(fia+ft)-i)>i.
q ly n\N ly \j=l J
n>\
by (2.5.3). This completes the proof of Lemma 2.11. □
We are now in a position to construct the \J/(ri) of Theorem 2.8. We do this
inductively using Lemma 2.11 as follows. Take Rl = 1, €l = \ to obtain a sequence
The Duffin and Schaeffer conjecture reformulated 53
ag(l), say. We then apply Lemma 2.11 again with R2=N(RU ej, €2 = i. In general
we take Rj=N(Rj_l, tj-i), €j=2~J" and obtain 0Lq(j). Put
Then 0 < \j/{n)<\ and
n=l
diverges. However, (2.1.14) has infinitely many solutions only if a belongs to
infinitely many sets with measure €7-. Since
oo
we conclude, by Lemma 1.2, that (2.1.14) has only finitely many solutions for
almost all a, which completes the proof. □
Of course, the sequences constructed by Lemma 2.11 satisfy
4* n n 2
so the example we have given is not a counterexample to the Duffin and Schaeffer
conjecture.
2.6 THE DUFFIN AND SCHAEFFER CONJECTURE
REFORMULATED
It will be useful to consider the following equivalent statement of the Duffin and
Schaeffer conjecture to examine more cases for which it is known to be valid.
Let an be a sequence of distinct positive integers, and \\/{n) a non-negative function
of an integer variable. Then if
I *<»>^ (2.6.1)
n=l an
diverges, there are infinitely many solutions to
\ccan-m\<il/(n), meZ, (m,an) = \, (2.6.2)
for almost all cceU.
We shall tackle this problem by reducing the task to the estimation of sums
of the form
I Am)/(.n) C^^t"
\ avPn
54 Diophantine approximation
for some a>0. In the next chapter we shall bound such sums involving common
divisors for the case o=\. Here we shall prove the following result.
Theorem 2.9 // an is a lacunary sequence, then the Duffin and Schaeffer
conjecture, that the divergence of (2.6.1) implies that there are infinitely many
solutions to (2.6.2), holds.
As usual, by 'lacunary' we mean that
>c
a
for some fixed c>\. We begin with the simple observation that we can assume
\l/(ri)^n~2 for all n, since this only alters the set of a for which (2.6.2) has infinitely
many solutions by a set of measure zero (by Theorem 2.4). Also, we may suppose
that \j/(n)^. To see this, note that at least one of
x «„)**>, i wo**)
^(n)>l/2 an MnKl/2 Qn
must diverge. If the former sum diverges, we discard those an for which \l/{n)^\
and relabel the remaining sequence. The factor P(m,ri) given by (2.4.2) is then
«1, since
and so we obtain l(£mc\£n)« X{S„)X{Sn). If the latter sum diverges, we can
replace \j/{n) by min(i^(«), ^).
We shall use Lemma 2.8 with the simplification of replacing D(m,n) by
( a a V'2
E(m,n)=(—^-Iil,(mMn)) ,
which is no larger than D(m,n) in view of the inequality (ab)1'2 <max(fl,6).
We write
L(m) = - log(iKm)), L2(m, n) = log( - log(^(m)^(«))),
Aim ii) ^^)2Um)m
aman \lf{m)\lf{n)
We also note that, for any integer r^2,
II (l-f) ^expf- I logd-p"1)
P\r \ PJ \ P\r
p>logr p>logr
The Duff in and Schaeffer conjecture reformulated 55
«exp(ll)<exp(^p)«l. (2.6.3)
p>logr
We shall use Aim, n) as an upper bound for the characteristic function of those
(ra,«)eZ2 for which P(m,ri) is large. The next lemma establishes the required
property of A(m, n).
Lemma 2.12 If
A(m,n)<\, (2.6.4)
then
X{SmnSn)«l{Sm)X{Sn). (2.6.5)
// (2.6.4) fails, then
l{Smnin)«l{£m)X{gn)L2{m,n) (2.6.6)
«X{Sm)k{Sn)L{m)L{ri). (2.6.7)
Proof If
E(m,n)> -log((flw,gw)2 ), (2.6.8)
in n
then (2.6.5) follows from Lemma 2.8 with (2.6.3). We may thus suppose that
(2.6.8) fails. It follows that
i>(m,H)«'0g'°g(3^"/(a^))2, (2.6.9)
max(l,£(m,«))
since, for Y < log X,
sH'W s ?) by(2-63)
p>Y Y<p<\ogX
^ ( i\ log log jt
< exp > - « ————.
\y<P<iogxPj log^
Now suppose that (2.6.4) fails. Then
gw,flw 2 ^ il/(m)~ V(")_1^(w)^(")
(flmaHy
^il/(my2\l/(ny2.
Hence (2.6.9) gives P(m,ri) « L2(m,ri) and so (2.6.6) follows from Lemma 2.8. Of
course, (2.6.7) is an immediate consequence of (2.6.6). It will be vital in our later
discussion that P(m,ri) is bounded by logarithms of \J/(m) and \J/(ri) (which are
themselves bounded below by m~2 and n~2 respectively), and not of am and an.
56 Diophantine approximation
Now suppose that (2.6.4) holds. We must distinguish two cases:
(i)If
anfln >bl/(m)il/(n)r2, (2.6.10)
(flmaH)
then
1/2
2^1 anfln \ '* ,_/ ^n
E{m,nY>\f m " ^log
It then follows from (2.6.9) that P(m,ri) « 1, which gives (2.6.5).
(ii) If (2.6.10) does not hold, then
log( . ^".J < -21ogOK™)<M"))
« L(m)L(n) < *"'*", ij/(n) \j/ (m) by (2.6.4)
= E(m,ri)2.
Hence P(m,ri) « 1 from (2.6.9), and this completes the proof. □
It follows from the above result that
I k(Smc>Sn) « ( £ ^«ii)Y +1, (2.6.11)
ls£»i<nsSN \n=l fln
where
I„ = I ,4(m, n)* ^ y (° »(/w)»(n)L2(m, w).
l<m<n<N fln^m
Here er is any positive number, and we are at liberty to replace L2(m,ri) by
L(m)L(ri) in the above. To prove Theorem 2.9, we take c=i, which gives
Zi/1= Z ((f=^!)I'4(i(m)L(„))5/.(Wm)W„))3,4^MO
>2\l/4
«
Z (%^V WW,r^, (2.6.12)
1<1
Now
l<m<n<N\ flmfln / *V*m
(am'an)2 <aJa<cMm-n)
The Duffin and Schaeffer conjecture reformulated 57
for some c>\, since the sequence is lacunary. For any non-negative function/(«)
we have
X f(m)f{n)cm-n ^\ X (f(m)2+An)2)cm-n
N oo 1 N
<lf(n)2Y,c-j=-—r Z/(*)2-
n=l j=0 C l n=l
Thus the sum in (2.6.12) is
as required to complete the proof. □
We finish this chapter with one further result. One question that naturally
arises out of our reformulation of the conjecture is: if we fix \j/(ri) initially, for what
sequences an are there infinitely many solutions to (2.6.2) for almost all a? The
following theorem and its corollary deal with one simple case.
Theorem 2.10 Suppose that \j/(ri) is a sequence of non-negative reals and an is a
sequence of distinct positive integers such that (2.6.1) diverges,
il/(n)^^- =0((«lognloglog«)_1), (2.6.13)
an
and
nA « an « nB
for some fixed A>0, B^\. Then there are infinitely many solutions to (2.6.2) for
almost all a.
Corollary The above result also holds with (2.6.13) replaced by ^(«) = («log n) ~l.
Remark Vaaler's theorem in the present notation is only \j/{n) = 0{a~l).
Proof of Corollary We do not increase the number of solutions for any a by
replacing \j/(n) with
1 minfl, a"
nlogn \ (p(an) log log 3n
The conditions of Theorem 2.10 are then fulfilled for this function. □
Proof of Theorem 2.10 Let 9 = A(4 + 2B)~A. We can assume as before that
\j/(n)>n~2. Now, if m<nd, we have
an^irn) fl>(m) nAm 2
^ » is— ^ n
All
58 Diophantine approximation
Hence, by Lemma 2.8, l{£mn£n)« A(<fm)A(<fn). It therefore suffices to
demonstrate that
£ ^wfW^togtogft.J^j'WY. (2.6.14)
However,
n=l anfln \n=l fln
n°<m<n
£ *(m)*^«c I , J , « l
nv<m<n
n=l
00
am n'<m<nmlo§wl°§lo§m k>glOg9«'
and so (2.6.14) is certainly true, which completes the proof. □
The question might be asked: can we not allow \j/{n) to be bigger and somehow
slow the divergence of (2.6.1)? In the next chapter we do slow the divergence with
\j/(ri) = 0(n~1) for certain cases (see Lemma 3.8). We have a problem here though,
since even if a sequence <5(«) is tending to zero monotonically, it is still possible for
I<5(")
to diverge, yet
£ min(«5(«),(«log«)_1)
n=l
converges. This cannot happen if nlogn is replaced by n.
NOTES
Walfisz [275] appears to have been the first to consider restricting the
denominators allowed in Khintchine-type results. He was interested in integers #2 (mod 4).
The Duffin and Schaeffer conjecture is the most important and stubborn problem
in this area. In the next chapter we shall consider further cases, but the proof of
the general result will still escape our grasp. Strauch [255]-[258] has considered
various cases, but our results supersede his. Theorem 2.10 in this chapter has not
previously been published. It forms part of a family of results that can be obtained
which have the form 'If we have some handle on the growth of \j/ and the an, we
can establish the result'. It would be nice to dispense with (2.6.13) and require
only the conditions nA « an « nB. It seems (see the next chapter) that the 'worst'
sequences may consist of numbers having only small prime factors.
Kargaev and Zhigljavsky [150] have investigated the function
F(a,N)= mm .
Notes 59
They deduce from Khintchine's theorem that, if/is an increasing function with
/(1)>0, then
has infinitely many solutions for almost all a if and only if
J i M
diverges. Also, they prove that N2F(a,N) ^/(logiV) has infinitely many solutions
for almost all a under the same conditions on/ This last result can also be deduced
from Khintchine's theorem by an elaboration of the proof of Theorem 2.3. In this
context we also mention the recent work of Burger [52] on the expression
2 „,_ Ml
v(a) = limsupA?r min
which is simply connected to the function F(a, iV) above. This number is finite
only on the set of a of measure zero for which
liminfiVlla/ill
N-oc
is positive, that is, the exceptional set for Theorem 2.3.
The theory of approximating any irrational a with fractions whose
denominators belong to a given infinite set has a long way to go before it reaches the
best-possible sharpness of the metric results we consider. For example, with
an=f(ri), where/is a polynomial of degree k with integral coefficients and positive
leading coefficient, we have
II«/(")!! <
«10g«
infinitely often for almost all a, but the best results known to date for any
irrational (see [18]), have the form
\W(n)\\<n-a(k\
where a(k)^0 as k increases. If we take an as the nth. prime, then
\\*p\\<p-1
infinitely often for almost all a, but the best result to date for all irrational a (unless
one assumes very strong results on primes in arithmetic progressions) is ([126])
\Wp\\<p-1122.
However, it has been shown [125] that there exist uncountably many a such that
n I. 1°&P
|| a/? ||» .
3
GCD sums with applications
Asymptotic formulae reduced to GCD sums. Bounds for GCD sums.
Extensions to higher dimensions. Applications to the Duffin-Schaeffer
problem. A generalization of Khintchine's theorem.
3.1 STATEMENT OF RESULTS
Sums involving common divisors were introduced in one context in the previous
chapter. The following result demonstrates how they arise in a related problem.
Throughout this chapter J5" denotes the set of real functions/ such that/(«)e [0, j)
for n ^ 1, and £*= x f(n) diverges.
Theorem 3.1 Suppose that fe 3P'. Let pn be a sequence of reals, and an a sequence
of positive integers. Write S(a, N) for the number of solutions to
llafl. + ftj </fo), n^N. (3.1.1)
Then, for almost all a,
S(<x,N) = ®(N) + 0(E(N)ll2(\ogE(N))3l2+*). (3.1.2)
Here,
n=l
and
Em- I (am,an)^mm. (3.1.3)
This result is a slightly refined version of something proved inter alia in
LeVeque's paper [180]. It immediately implies his Theorem 1, which we state in
an improved form as a Corollary.
Statement of results 61
Corollary Suppose that an is an increasing sequence of integers and fe^. Let f$r
be a sequence of reals. Then, if
&^&M«-wm™ (3-M)
for some <5 > 0, we have, for almost all <x,
S(ol,N)^®(N) as TV-► oo.
The sum on the left-hand side of (3.1.4) can be explicitly evaluated in some
situations. For example, if/(«)=«_1 and an = n, then
£(A0 = C(logJV)2
for a certain constant C, so we just fail to obtain an asymptotic formula in this
situation. We shall need to use a different approach in the next chapter to deal
with this problem. On the other hand, if an=nk for some integer k ^ 2, then
X (am,an)^nk lo(n),
m=l
where
a{n) = YJd.
d\n
Since a(n) has average value nn2/6 (Theorem 324 in [111]; compare Lemma 2.4
above), it follows that for /(«) non-increasing,
E(N)«®(N),
and so we obtain an asymptotic formula. We also obtain such a formula when
an = kn for some integer k ^ 2, or if an is a subsequence of the primes.
In more general situations we can use the fact that
1 " 1 a"
— Z ^m,an)^- X (m,an)^T(an).
anm=l anm=l
We then need to prove (essentially) that
N
Z/(«)T(fln)«0(AT):
n=l
If an grows rapidly and we have no information on the number of divisors of an,
this bound is of little use. However, if an is lacunary, then
1 " 1 "
- Z («»»«.) <t Z *m«l>
anm=l anm=l
which gives an asymptotic formula in this case (Theorem 7 in [214]; Philipp
indicates there that a law of the iterated logarithm may hold in certain cases). In
view of Theorem 2.8, the error in a formula like (3.1.2) must sometimes dominate
the main term.
62 GCD sums with applications
We now write
w(r,s) = \
aman
Using the inequality min(a, b) ^ (ab)112, we then obtain
E(N)^ X w(r,5)(/(r)/(^))1/2. (3.1.5)
r,ss£N
We have thus obtained an upper bound for E(N) in terms of the type of sum
introduced previously (£, in the notation of Chapter 2).
In the 1930s, J.F. Koksma realized that bounds for the sum
Z ™2(r,s)
r,s*S:N
would have implications for the distribution of the sequence anoc modulo 1 for
almost all a. Although easy to bound for many sequences of interest, the maximal
order of the sum in general proved to be a harder problem. This was determined
by I.S. Gal [100] to be «JV(loglog9iV)2, a bound which is genuinely attained for
certain sequences. The sum we study in (3.1.5) was first considered in [79]. The
present author with T. Dyer there obtained the following result.
Theorem 3.2 Let f and g be non-negative real-valued functions and ar a sequence
of distinct positive integers. Then, for all N^l, we have
,j„<*,i"'),""*a<<(,i«,*"",""<SS)))"! <316>
where C is an absolute constant which does not exceed 5.
We shall show in Section 3.3 that the factor
/ Clog/2
6XPVloglog(9«)/
cannot be replaced by a factor smaller than
expi^i^oJ' (3'L8)
It is tempting to conjecture that the factor in (3.1.8) gives the 'correct' maximal
order of magnitude. In any case, this factor sets a limit for the method we are
using. It will be easy to deduce the following result [116] from Theorem 3.2.
Statement of results 63
Theorem 3.3 Let the hypotheses of Theorem 3.1 be given. Then, for almost all a,
we have
S(a,A0 = <D(A0 + O(0>l/2(A0), (3.1.9)
where
^N)=km^n)Aw^)Aios2n)i} (3uo)
We note that if O(JV) »Nd for some 5 > 0, then, for almost all a,
S(<x,N) = <l>(N) + 0(<bm+\N)) (3.1.11)
for any € > 0. We pause to make a brief comment on the € > 0 in some of these
theorems. Usually € > 0 is fixed at the beginning and everything in the conclusion
might depend on €. For these results the set of 'almost all a' can be made
independent of e, so that the formula (3.1.11) is true for the same set of almost all a
for every € > 0: only the implied constant depends on €. To see this, let £(€) be the
exceptional set for which the formula is not valid. Then put
Clearly A(<f ) = 0, being a countable union of sets of measure zero. If cl$£ , it
follows that (3.1.11) is true for every e > 0.
Theorem 3.3 improves the result of LeVeque [182], who had an error term
JV1/2(log jV)3/2+€ in place of our 0>X{N). He conjectured that given y 6(0,1/2), one
could find sequences an such that the asymptotic formula is invalid for/(«)=ny~l,
but Theorem 3.3 disproves this. Results by Schmidt which are sharper but less
general will be discussed in the next chapter.
We now consider a higher-dimensional analogue of Theorem 3.3. We write ^
for the set of functions /(«) with/(/i)/2e^, and denote by A the integer lattice
inR*.
Theorem 3.4 Let if/^n),. ..,\j/k(n) (k^2) belong to ^i, and suppose that the
function
k
<K")=n<A»
also belongs to &x. Let a sequence of distinct integers ar and a sequence of
boxes @t (r) a Uk be given, where the sides of each box are parallel to the coordinate
axes and of length if/1(r),. ..,if/k(r). Then, for almost all ae(Rfc, the number of
solutions to
anai + me(%(ri), meA, l^n^N,
(3.1.12)
64 GCD sums with applications
is
^(iV) + 0(vFl/2(A^)(logvF1(iV))3/2+€), (3.1.13)
for every € > 0, where
Here, for k = 2,
^(«) = iA(«)Klog2«)4+€ + (iA1(«) + iA2(«))exp
while, for k^3,
<5(H) = lA(w)(l + (l0gw)4+€ Z fl^O
To clarify the situation, which may be obscured by the clumsy form of the
expression <5(«), rearrange \J/X(r),..., \J/k(r) as \l/f (r),..., i^jf (r) in increasing order
of size. We then have, for k ^ 3,
d{n)«\J/(n)(\ + (log2«)4+€ \\f% (n)
+ ^(„)^Wexp(^g;)).
It follows that in any genuinely three-dimensional (or higher-dimensional)
situation the asymptotic formula holds. Clearly there must be some restriction on the
shape of the boxes, otherwise the problem could degenerate to the one-dimensional
case, for which we know the formula can fail.
Theorem 3.4 improves the work of Gallagher [106] and Ennola [86] on this
problem. Gallagher dealt with cubes for k ^ 3, and Ennola with boxes whose sides
were in bounded ratio. Our result allows the sides to be of quite different sizes,
and gives a much better error term in the cases for which their theorems are
valid: O(A01/2(log<E>(JV))3/2+€ rather than Gallagher's d>(A/y+€-1/(2fc+2) 0r
Ennola'sO>(JV)1+€-1/(2k).
We can generalize the situation further by replacing boxes aligned with the
coordinate axes by arbitrary convex bodies. Not surprisingly, our result in this
case is not so precise as above. We first need some notation to describe the bodies.
By the diameter of a body we mean the length of the longest line segment which
/ 51ogK \\
Vloglog(9«)i/
Statement of results 65
lies wholly within the body. Let a sequence of convex bodies Mir) c Uk be given.
For 1 ^j^k — 1 let Aj(r) be the maximal k —y-dimensional volume of the
projections of £8 (r) onto the coordinate hyperplanes of dimension k —j. We put Ak(r) = 1
for convenience, and write V(r) for the volume of @(r).
Theorem 3.5 Let k, an integer not less than 1, a sequence of distinct integers an, a
sequence of convex bodies 31 (r) c Uk each of diameter less than 1 and the sum of
whose volumes is infinite, and e > 0 be given. Write
0>(AO= £ V(n).
Then, for almost all a e Uk, the number of solutions to
ancn + me@(ri), me\, l^n^N,
is
where
O(A0 + O(A1/2(A0(logA(A0)3/2+€), (3.1.14)
A(A0=I minCKW^l)
n=l
and
^^(loglog^^^^^-^exp^j^^
+ ^(21/2)-€(«)(log2«)4+€. (3.1.15)
Moreover, ifk^3 and, for all n and some <5 > 0, we have
A! (n)« exp( * j and A2(n)«(log In) ~ 8 "',
then we can replace (3.1.15) with
£n = (loglog9«)1+€.
For our final applications of Theorem 3.2, we reconsider the Duffin and
Schaeffer conjecture.
Theorem 3.6 The Duffin and Schaeffer conjecture is true in k dimensions, where
k ^ 2. Indeed, let \J/ 1(n),..., \l/k(n) be functions ofn taking values in [0, C) for some
C>0. Write
k
0(n)=U ij/j(n),
j=i
and suppose, for some positive reals € and K, that for each n for which 9(n) ^ 0 we
have
max ^-<K6{nf. (3.1.16)
66 GCD sums with applications
Let an be a sequence of distinct positive integers for which
„=l\ an J
diverges. Then, for almost all a = (a1?..., <xk)eUk, there are infinitely many
solutions to
\an0Lj-rj\<\lfs(n)9 (rj9aH) = \, r}eZ, (y = l,. ..,£). (3.1.17)
The ^-dimensional form of the Duffin and Schaeffer conjecture was first
proved by Pollington and Vaughan [225]. They proved the result with
\j/j(ri) = 9(n)llk for l^j^k. Their method, which was based on the ideas of Erdos
that we used in Section 2.4, could give a slightly more general result, but appears
to require the situation to be at least two-dimensional. The result we prove here
(which was proved in [122]) can be interpreted as saying that the situation must
be at least 'l + e'-dimensional. This is the significance of (3.1.16). Pollington and
Vaughan do allow \J/j(ri) to be unbounded; this could be permitted here, but the
hypothesis (3.1.16) would become more complicated.
Theorem 3.7 The Duffin and Schaeffer conjecture holds, given any of the
following conditions:
(i) an is a subsequence of the kth powers for some k ^2;
(ii) q>(an)lan<Kn~d for some K, 5 >0;
(iii) for infinitely many N,
n— l n
for some c, S > 0;
(iv) \j/(n)>c[^^-) for some c,R>0;
,6-1
(v) a„+! Ian > 1 + cnb x for some c, b > 0, if either
(a)^Wn)< , Kf' Q for some K,
an log«loglog9«
or
(b) \J/(n) is non-increasing and
( I Kn* \\ c
an« exp exp -—-——-—— for some K.
Although many situations are covered by the various partial solutions that we
have given to the Duffin and Schaeffer conjecture (Theorems 2.5,2.6,2.9, and 3.7),
we still appear to be some way from proving the conjecture in its generality.
Theorem 3.7 appears in [122]. In the final section of this chapter we give a proof
of the following analogue of Theorem 3.6 with the condition (j,an) = \ removed.
Proof of Theorem 3.1 67
Theorem 3.8 Given the hypotheses of Theorem 3.4, except that the boxes are
centred on the origin, there are infinitely many solutions to
ana + me@(n), me\, neZ, (3.1.18)
for almost all cneMk provided that (3.1.16) holds for some e>0.
This result generalizes the work of Gallagher [106], who proved this result for
\J/j(ri) = \l/(n)1,k. In his theorem the boxes are all in one quadrant with one corner
at the origin, but there are no problems in modifying our proof to cover this
situation. For a different generalization of Khintchine's theorem to k dimensions,
see Theorem 4.6 below. Of course, Theorem 3.8 follows immediately from
Theorem 3.4 when
max ..... <<*j/(ny.
\J/r(n)\J/s(n)
In other words, whereas Theorem 3.4 requires a genuinely three-dimensional
problem to give an asymptotic formula in all cases, Theorem 3.8 will give infinitely
many solutions in a two-dimensional case. Theorem 3.8 gives a very general
extension of Theorem 2.2 (Khintchine's theorem) to simultaneous approximation,
with only a mild shape restriction. Theorem 4.6 below removes the shape
restriction for boxes (indeed, more general shapes are allowed) but imposes: an =
n, \j/(ri) is non-increasing.
3.2 PROOF OF THEOREM 3.1
We first of all prove a general lemma which will be useful for several of the results.
We write U for the unit cube in k dimensions. Given a set s4 <= Uk, we write, for
/?>0, f$<srf for the set {aeRk: aj9_16j^}, and N(j&) for the number of lattice
points in stf.
Lemma 3.1 Let 3F, <§ c Uk be two bounded measurable sets, with f and g their
respective characteristic functions. Write
F(x)= X /(Jt + m), G(x)= X gix + m).
Then, for any two positive integers a, b we have
r
F(ax)G(bx)dx=——k
u (aft
1 " N(a$-y)dy (3.2.1)
1
/•
wj
N(p3?-y)dy, (3.2.2)
68
GCD sums with applications
where
a
cc =
(a, bV
P =
(a,b)
Proof Write A* = {jteA:0^xj<a(y' = l,...,«)}. Then we note that the
expression ny + can runs over all the vectors in A exactly once as n runs through A* and
m through A, provided that yeZ with (y,a) = l. We shall use this fact twice in
the following, first with y = 1, and again at the end of the proof with y = p.
Since the integrand on the left-hand side of (3.2.1) is periodic in each coordinate
with period (a, by1, we have
F(ax)G(bx)dx =
u
F(ax)G{Px)dx
u
= 1 I
me\ neA*
f{<xx+n + am) G(px) dx
u
(since/and g are non-negative measurable functions, we are justified in swapping
summation and integration)
-z
neA*
R*
f(ouc+n)G(Px)dx,
since G(px) = G(p(x+m)) for any we A,
1
= 1
«6A*
/ijW*?*W
with the change of variable y = <xpx + /to
1
(aft* J
ZG(rf)d,.
^ ne A*
a
(3.2.3)
The proof of (3.2.1) is completed by noting that the integrand in (3.2.3) counts
the number of solutions to
mcc+nfieoiy—y, weA*, meA,
and this number is N(<x<&—y) by the remarks at the start of the proof. Clearly
(3.2.2) follows similarly. □
Now, in R* it is evident that for every interval J we have
N{pS-y) = pX{S) + 0(l).
If we let J^and ^ be the sets of xe[0,1) for which
H*+iM<yi and Il*+02ll<y2
Proof of Theorem 3.1
69
respectively, we thus obtain
A({x:||ox+/y<y1 and ||fo + jU <y2}n[0,l))
F(ax)G(bx)dx=— .
o a^J
1
N(*<g-y)dy
p&
(3.2.4)
= — X{p3F){X{oL<&) + 0{\))
= 4y1y2 + 0(y1(a,&)<r1).
Using (3.2.2) in place of (3.2.1) gives an alternative bound
4y1y2 + 0(y2(a,W1).
Hence, if £} is the set of a e[0,1) with || aaj+PjW <f(j), we obtain
X(£jn£k)=4f(j)f(k) + (aj,ak)mm(^,^
\ aj ak
We remark that LeVeque obtained (3.2.5) using the Fourier series expansion of
the characteristic function of an interval (mod 1).
Now let /(a, n) be the characteristic function of Sn. We then have
l' I (/(a,«)-2/(«))Yda
(3.2.5)
0 \r<n<s
1*1
1*1
X /(a,«)/(a,m)da-4 £ f(m) £ /(<*,«) da
0 r<n,m<s r<m^s JO r<n<s
+ 4 Z f(m)f{n)
r^m,n^s
= X Wnn<fJ-4/(m)/(«))
I since
/(a,«)da = 2/(«), and
/(a, «)/(a,
m)da = Mnn*m)
« I (an,am)min(-—,—— 1
by (3.2.5). The formula (3.1.2) then follows immediately from Lemma 1.5. □
We finish this section with a slight refinement of (3.2.5). Suppose that
A(P&r)>\. Then /fJ^can be divided into [HP^)] intervals of length 1 and an
interval of length {X(p!F)}. Now the integral of N(ot&—y) over any interval of
length 1 is X{ol^). Hence
p&
N(ol% -y) <\y = [X(P^)] X(ol<& ) + {X(P^)} (A(a^) + O(l))
= X(aL<g)X{fS&) + 0{\).
70 GCD sums with applications
We can therefore improve (3.2.5) to
AC^n^=4/OV(A:) + 0(^ (3.2.6)
V \aj ak ajak ))
In practice this is significant only when f(n) decreases slowly.
3.3 PROOF OF THEOREM 3.2
We first state a result which is equivalent to Theorem 3.2. The deduction of the
following result from Theorem 3.2 is relatively easy, but the converse deduction
is more complicated. After thus establishing the equivalence of Theorems 3.2 and
3.9, we shall need to prove some lemmas before eventually proving Theorem 3.9.
Theorem 3.9 Let M,N, R, S be positive integers and let {ar}, {bs} be two sequences
of distinct non-zero integers. Then the number of solutions to
mar=nbs,
with
l^m^M, Kw^JV, l^r^tf, l^s^S, (3.3.1)
is
where C is an absolute constant less than 5.
Remark Since the implied constant is absolute, the word 'integers' could be
replaced by 'rationals' without altering the conclusion (simply multiply each of ar,bs
by the highest common denominator for l^r^^R,l^s^S). Moreover, the word
'integers' can be replaced by 'reals' since we can find disjoint sets &j9 Sf i such that
ar eM^ bseSfk=>arlbse Q if and only if j = k.
The result may then be deduced from the corresponding result for rationals via
Cauchy's inequality. For real sequences, however, the more interesting question
involves
\arm—bsn\<6.
We shall consider this in Chapter 7.
The reader should have no trouble in deducing Theorem 3.9 from Theorem 3.2,
since, for given values of r, s, the number of solutions to mar=nbs with l^m^M
and l^rt^AMs
mm
M
-r(ar,bs)
~(ar,bs)
L.ar
K(M7V) (*A)1/2"
Proof of Theorem 3.2 71
We then apply Theorem 3.2 with cn as the sequence of all integers up to the
maximum of ar and bs with
fl if n=ar for some r^R
[0 otherwise,
fl if n=bs for some s^S
g(n) = { f
[0 otherwise.
We now deduce Theorem 3.2 from Theorem 3.9. For convenience we write
We first note that there exist real numbers R,S^N (we assume that iV>4) such
that
I (/(r)^))1/2w(r,5)«(log3JR5)3 £ (f(r)g(s))ll2w(r,s), (3.3.2)
lsgr.s^N r~R
s~S
where a~A means A^a<2A. Now let
F=max/(r), G=maxg(s),
r~R s~S
and note that the values of r or s with f(r)^F/(RS)2 or g(s)^G/(RS)2 can be
discarded, as their total contribution to the right-hand side of (3.3.2) is at most
(FG)ll2(\ogRS)3«( X f(r)g(s)X'\\ogRS)3,
s~S
which we shall see is a suitable estimate. We may then split up the remaining
values of r and s into sets @(f),Sf(g), respectively, so that f(r)~f for re@(f)
with/taking on «(log.RS) values, and similarly for g(s)~g when geSf. Hence,
for one pair /, g, we have
I (f{r)g{s))li2w(riS)«{\ogRS)5 £ (fg)1/2w(r,s). (3.3.3)
lsgr.sssN re£(f)
Now let X be the number of solutions to mar=nas with re<%(f),seSf{g),
l^mn^RS. Since a solution has m=has(ar, as)~x, n=har{ar, as)~l for some
positive integer h, we have h2^RS(ar,as)2/(aras), so that
x= ^ Ri^)1/2(y<)1=(Jgst)1/2 ^ W(r,s) + 0(|a(/)||*>(0)|). (3.3.4)
r e#(.f) L W- fls) J rem/)
se^ig) seSf(g)
72
GCD sums with applications
By applying Theorem 3.9«(\ogRS) times, we obtain
N<<{RS\a{my(9)\)1,2E(iC,IW<\ogRS). (3.3.5)
Combining (3.3.3), (3.3.4), and (3.3.5) then gives that
Z (/(r)^))1/2w(r,5)«(log^)6(|^(/)||^(^)|/^)1/2JE:(iC,^)
«(\@(f)\\y(g)\fg E(C',RS))112
«( I fir)g{s)E{C\rs)
where C' = (5 + C)/2<5. This establishes (3.1.6) as desired. The bound (3.1.7)
follows, since
E(C, AB)« E(C, A)E{C, B).
We have thus reduced the proof of Theorem 3.2 to the proof of Theorem 3.9.
To prove Theorem 3.9 it clearly suffices to establish the result when both
sequences are positive. We now explain the basic idea behind the proof. Suppose
we were considering the equation
arm=asn with \^r,s^R, \^m,n^N, (3.3.6)
and knew that the integers ar were of the form Huh with \^u,v^Rc for some
fixed c. It would then be straightforward to bound the number of solutions to
(3.3.6) as «NR1 +€. It would therefore be desirable to split s4 = {ar: l^r^R} into
subsets s/j such that if ases^^ then as=aru/v for some r = r(j) and \^u,v^Rc.
We should also like arm=asn to imply that both ar and as belonged to the same
jrfj. Finally, stf ought to be the disjoint union of the .stfj. Since this set-up is
impossible, we begin by constructing pairs of sets &jt @j such that #,-sQ.p
I^jIkR'I&jI (so Q)j is not too much bigger than &j) and s& is the disjoint union
of the &j. We also require ase2ij to imply that as=aru/v for r = r(j) and
l^u,v^Rc, and, finally, if ars^j and arm=asn, then ase@j. This gives a
bound «NR1 + 2€ for the number of solutions to (3.3.6). Following this approach
gives a result with E(C/2,RS) replaced by
/ K log RS \
eXPV(loglog9^)1/V'
if one replaces R* by the maximum of the divisor function. We need to work more
carefully in order to obtain our result, but the principle remains the same.
We note in passing that the problem considered by Theorem 3.9 is very natural
in the context of overlap estimates, since it essentially counts the coincidence of
centres of the subintervals making up Sr and Ss.
We start by obtaining the best bounds for the number of solutions to the
equation arm=asn when some bound is given on the size of ar,bs.
1/2
Proof of Theorem 3.2 73
Lemma 3.2 Suppose that r,N^\. Then
XT(")r^iV(log3iV)2r-1. (3.3.7)
ns$N
Corollary Let r,N,A^\ be given. Then
X \^N(\og3N)2r-1A-f. (3.3.8)
Proof Write LS(N) for the number of points in Zs with positive coordinates on
the hyperplanes x1...xs=n, n^N. We have
T(«)r^T2,(w),
where zy{n) denotes the number of ways of expressing n as the product of y
divisors. Thus the left-hand side of (3.3.7) is bounded above by L2r(N). Since, for
iV>l, we have
L.iN^N, Ls+l(N)= X Ls(N/h), and £ \ <log3iV,
the bound Ls(A0 < N\og~1 3Nfollows by induction. Hence (3.3.7) is established. □
Of course, precise asymptotic formulae are known for the left-hand side of
(3.3.7), but we require a simple bound with no error terms involving r implicitly,
since we shall take r growing with our main parameters. The bound (3.3.8) follows
immediately from (3.3.7), of course.
Lemma 3.3 Let r,n^l. Then we have
l(T-^\<<E(2W0r),n). (3.3.9)
Proof Since x(n) is multiplicative, we have
<
H(i+^rVd/,)-ji(i+sf!
«£(21/(10r),>2). □
74 GCD sums with applications
Here the final inequality was obtained by applying standard procedures to the
preceding product (see Chapter 18 of [111]). The number 21/10 arises as 2 + e, but
the inclusion of an € would complicate the details later.
Henceforth (w, n) may represent a point in Z 2 at times, but it should be clear
from the context when this happens and when (m, n) denotes the greatest common
divisor. Given a subset £ <= Z2, we put
£(d,e) = {(m,n)e £: d|m,e|«}.
Lemma 3.4 Let £ c Z2, and suppose that there are positive integers M,N, such
that, for all (w,n)e£, we have l^m^M, l^w^JV. Then, for r^ 1, we have
X de\£{die)\2<<\£\MNE(l\l(5r\ii)4r(\ogii)2,*\ (3.3.10)
(d.e)eZ2
where n = 3 max(M, N).
Proof At the expense of a factor (log/*)4 we may assume that
P^T(m)<2P, Q^x(n) ^2Q
for (m,n)e£. Under this assumption we have
X de\£(d,e)\2 = X ^de £ 1,
,e)el2
and the innermost sum is at most
(d,e)eZ2 (m^nJeS d\ml {.mI,nI)eS'
e\nl d\m1,e\n1
1 I 1 . (3.3.11)
d\m, z(m)^P e\n,z(n)^Q
Now, if d\a, then T{a)^T{d)x{ald), so (3.3.8) gives an estimate for (3.3.11) which is
«^ri^f)Y(l0g,)2-2^ ^mt(^)y r,-2
We then obtain (3.3.10) by a double application of (3.3.9). □
Lemma 3.5 Let Hl,H2,T'^\, and suppose that £ l, £ 2 are sets of distinct coprime
integer points such that \^m,n^T whenever (w,n)e£1 or £2. 77ze« Me number of
solutions to
m1n2h1=m2n1h2 (3.3.12)
with (Wj, nj) e £j and 1 ^ /*,- ^ //,- (y = 1, 2) is
«(K1|K2|i/1/f2)-exp(^^_). (3.3.13)
Proof of Theorem 3.2 75
Proof We may suppose that m} ~ Mi and nj~Nj, and incur only the penalty of
a (log(3r))4 factor. Here 1 ^Mj,Nj^T. Now if hlyh2 satisfy (3.3.12), then
h1= — -, and h2= 1 2
(m1«2,w2«1) (/Wi«2»/W2/Ii)
for some integer /z. Employing the inequality min(a, b) ^ (ab)112 produces an upper
bound for the number of solutions to (3.3.12) which is at most
{HH)1I2 z f("^>"W2V/2
mj^eSjK mlm2n1n2
^H.H^2 £ de £ (m^y112 I ("W"1'2
( h h \1/2/ \1/2/ v/2
^\m nJn I «fc|*iW,«)l2 I de\S2{d,e)\2
«(//1//2K1|K2|)1/2£(21/(5r),r)4-(log3D2'+1 + 2
by Lemma 3.4. We obtain (3.3.13) by choosing r = [(10 loglog(9r))/7], whereupon
we have
£(21/(5r), r)4r(log 3T)2'*'+ 2
148 log 37
« exp 4log4loglog(9r) + 2(log9T)10(log2)/7 loglog(3r) +
«i
^..„e .„0„0_ , . v„0,- , ~„~0x~ , • 50(ioglog(9D)2,
since 10 log 2 < 7. D
iVoo/ 0/ Theorem 3.9 First we demonstrate that it suffices to show that the
number of solutions to
arm = bsn, l^mn^RS, l^r^R, l^s^S (3.3.13)
is
«RSE(C/2,RS). (3.3.14)
To see this, we note that the number of solutions with 1 ^ra^M, 1 ^n^N, is
X mm{[M{bsiar)b;'l[N{bs,ar)a;'])
lsgssSS
^MNRS)ll2 + l — \ X [(^/flA)1^^,)] 0.3.15)
\ / 1 ^r^K
76 GCD sums with applications
and the sum on the right-hand side of (3.3.15) is precisely the number of solutions
with l^mn^RS. Replacing this sum by the bound (3.3.14) then produces (3.3.1)
as required.
Henceforth we therefore suppose that 1 ^mn^RS, and write
j/0 = {ar:\^r^R}, @0 = {bs:\^s^S}.
We employ an inductive argument to split the above two sets into subsets so that
Lemma 3.5 may be applied and conditions corresponding to those given in the
sketch of the proof are satisfied. Now let
Fory^l the inductive step is as follows. We put
If either j&j or ^ is empty, the process stops. Otherwise, let ah be any member
of s/j. Then write
s#j(X) = {avesty. avm=ahn,\^mn^X,(m,n) = l},
@j(X) = {bve08j\ bvm=ahn,\^mn^X,(m,n) = \}.
Of course, s/j(X) c s/j(Y) for Y ^ X, and similarly for &j(X). Let y be a positive
parameter, which we shall choose optimally later. Clearly, for some integer
t ^ 1 + (loglog(9i?5))/y, we must have
(|^J.((^)t+1)| + l)(|^-((^)t+1)| + l)<
(|^((/wy)| + l)(|^((JW)')| + l) " ^ h
This corresponds, in the sketch of the proof, to finding 8p@j with ^.c Q)^ but
Q)} not too much bigger than &j. Now put
^'j=s/j((Rsy\ a'j=aj((Rsy),
This inductive process must terminate after at most R steps. Now if arm=bsn,
with are$4'p bse&j9 then we must have bse&j. On the other hand, \ib£0&p there
exists i<j such that bse&'i9 are&\. The number of solutions to (3.3.13) is thus at
most
KAWHtfiO')). (3.3.16)
j
where A^O') counts solutions with ares/pbse^j, and -/V2(y) counts those with
areJ^,6se^-. By Lemma 3.5 the expression (3.3.16) is
«(/JS)"2X((|^||^|)"2 + (|^||^.|)"2)exp(g5^g^);
Proof of Theorem 3.2 77
where r=exp((2 + (loglog(9ftS))/y)logC&S)). By our construction of the sets s/'j,
&j,&j,&j, we have, after an application of the Cauchy-Schwarz inequality, that
j
Choosing y = 61/2 then gives that the sum in (3.3.16) is
«RSE(C/2,RS),
where C=61/2 + 5/2 < 5, as required. This completes the proof of Theorem 3.9. The
reader will note that one could take C as (321og2)1/2 + € for any €>0 by working
more carefully. □
We finish this section by showing that the E(C/2,RS) factor cannot to reduced
below
(KQogRS)1'2}
eXPVloglog(9JRS)/
To do this, we need information on numbers with small prime factors. It is
reasonable to suppose that the 'worst' sequences for Theorem 3.9 will comprise
numbers having many small prime factors to high powers. Let ^(H^R) denote
the number of integers less than or equal to H, all of whose prime factors are at
most R. Hildebrand and Tenenbaum [134] have shown that, provided that
R(\og H) ~1 -*■ oo, we have
w,„ m „ ^^(a,/?)(27r(logi/)/log^)-1
*(H'R)= log(*/logtf) '
where
and a is the unique positive number with
p<rP l
We shall take H=exp(Rfi\ogR) with /?e[l/4,3/4] to be chosen optimally later,
and R an integer exceeding 2. We then have that
-I-'-'Sr)
< r^ (3.3.17)
for some absolute constant B (see Lemma 13 of [134]). By considering log^(a, R),
we then obtain
M>>og* + loglog9*Hw bgte
78 GCD sums with applications
and so xV{HiR)»H\ Now let
s# = {neZ: pr\n=>p<r and r^R2}, k = \j&\,
and write P for the number of primes less than R (so P = RilogR)'1). Then we
have
k = R2P, \ogk = 2P\ogR^2R.
Now suppose that m,n are integers with \^m,n^H and having all their prime
factors no greater than R. Then pr\m or pr\n implies that r^(\og2R)Rfi- The
number of solutions to
arm=asn, ar,ases/, (3.3.18)
is thus at least
(R2 - 2R* log2 R)p = R2P(\ - IR* ~ 2 log2 R)p » R2P,
since 2PRP~2 \og2R^>0 as R -»• oo. The number of solutions to (3.3.18) with m and
n running over all possible values is thus
»R2Px¥2(H,R)»HkH2a-\
We thus want to maximize H2a~x =exp((2a — l)RfilogR) as a function of k. Since
k is defined solely in terms of R, and in view of (3.3.17), we choose
1 loglog9i? Qg + 1)
P 2 logR \ogR '
We then have
/e-(B+l)nl/2\
HkH2" ~l^Hk exp —-—-—
__, /^(logft)1^
where # is an absolute constant. Hence, if we take M=N=H, R=S=k, and
br=ar as constructed above, we obtain the claimed lower bound.
3.4 PROOF OF THEOREMS 3.3 AND 3.4
Let c (ri) = (£(C, n)) ~l, and write
s# = {n: fin) <c(n)}, @ = {n: f{n)^c(n)}.
We shall suppose that both of the sums
£/(«) and £/(«)
Proof of Theorems 3.3 and 3.4 79
diverge, for otherwise the proof could be simplified. By Theorems 3.1 and 3.2, for
almost all a we have
I 1=1 /(«) + O(0>^2(iV)(loga>2(iV))2), (3.4.1)
nes/,n^N nes/
\\aan + p\\<f(n) n^N
where
®2(N)=YJf(n)/c(n).
nes/
Now, if h(r) e [0,1] for all r, it follows that
£/Kr)(log*( X *(*)Yf « I /K>0(log3r)6,
where log*x=max(l, log x). Since
0og3r)«cW-'«exp(di^),
we deduce that
02(iV)1/2(log<I>2(iV))2«a)1(iVr). (3.4.2)
Now, if we use the bound
W,.n^)-4/(y)/(/c)« ^^,
aiak
which follows from (3.2.6), we may apply Lemma 1.5 to deduce that, for almost
all a,
Z 1=1 /(«) + 0(03(^)1/2(log03(^))7/4), (3.4.3)
\\<xan + fi\\<f(n) n^N
where
03W= I (loglog9«)2.
To produce this formula we have appealed to Gal's result [100] that
(aj^ <<(r_M)(loglog(9(i;_M)))25
^'•"7^ CL-CLi
80 GCD sums with applications
where the implied constant is absolute. Since
O3(7V)1/2(logO3(A0)7/4<<( I (log«)4)1/2«01(A^),
combining (3.4.1), (3.4.2), and (3.4.3) produces (3.1.9) as desired. □
Before proving Theorem 3.4, we require the following lemma.
Lemma 3.6 We have, for all positive integers u, and any € >0,
I /(r)/(5)>v(r,5)2«X/2('')aog3r)4+€; (3.4.4)
if j3>3/2, then
I (/(r)/(5)w(r,5)V«Z/2^). (3.4.5)
Proof Since the ar are distinct, for any r we have
I w(r,s)2^ Z imny'
ss£S ls£m. ns£S2
«
f(log3S)2 if 0 = 1
(3.4.6)
Since/(r)/($K/(r)2+/(s)2, the result (3.4.5) follows immediately.
To prove (3.4.4), write
F(tf) = max/(r),
WR) = {F,m2-J:j = 0,l,...9l + [\og2R]},
sf(F,R) = {r:f(r)~iF,r~iR}.
It is not difficult to show that the contribution to the left-hand side of (3.4.4) from
pairs r,s for which one or both of r,s do not belong to one of the sets s/(F, R),
with F<^^(R), is less than some constant times the right-hand side of (3.4.4). It
thus suffices to consider
lire I %^,
R,S Fe&(R) re.s/(FyR) urus
Ge&(S) se^(G,S)
Proof of Theorems 3.3 and 3.4 81
with R and S taking on values 2,4,... ,2[l08jM]+1. Using (3.4.6), this sum is
« I I FGmm(\^(F,R)\\og2S,\^(G,S)\\og2R)
R,S Fe^(R)
^felogtf X F\s/(F,R)\1'2)2
(using min(a, b) ^ (ab)1'2)
by the Cauchy-Schwarz inequality
«£(log*)4+€ X F2|^(/s*)|
R Fe^(R)
« I /(r)2(log3r)4+<,
as required. D
Now let <% be the unit cube in Uk given by 0^Xj^\(j = \,...,k). It then
suffices to prove Theorem 3.4 for almost all xe^U. We put
fl if arxe@(r) + m,meZk,
r [0 otherwise.
We shall be appealing to Lemma 1.5, so we wish to consider
I (&,(*)-*(*))) dx
xe& \r^n<s /
X gn(x)gm(x)dx-( X iA(«)Y. (3.4.7)
X6*r<n,m<s \rsSn<S /
Since the boxes ^(r) are aligned with sides parallel to the coordinate axes, we have
k
9n(x)= EI hl(Xj> (* = (*1» • • >>Xk))>
82 GCD sums with applications
where the functions h{{xj) are characteristic functions of intervals (mod 1). Thus
k fl
hjn(x)hjm(x)dx
0
X gn(x)gm(x)dx= X n
xe^ r^n,m<s r<n,m<s;=l J
= Z Ei Uj(n)il/j(m) + 0((ij/j(n)ij/j(m))ll2w(m,n))
r<n,m<s;=l \
by working as we did in Section 3.2.
Now write
jaf = {a = (a1,...,ak)eRk: a,=0 or i,a#0}.
We can therefore bound (3.4.7) by some constant times
III (woW/n))1""'w(m, «)2aA
ae.o/ r<n,m<s j= 1 \ /
and apply Theorem 3.2 and Lemma 3.6 to the appropriate terms. For example,
(3.4.4) is applied to each of the kC2 sums with £j a, = l to obtain a bound
« X dog")4+€ *(*) £ II *»(")•
We thus obtain that (3.4.7) is «*F1(^— 1) — ^(r — 1), which completes the proof
by Lemma 1.5. □
3.5 PROOF OF THEOREM 3.5
We recall that for Theorem 3.5 we wrote Aj for the maximal k —y-dimensional
volume of the projections of a convex body onto the coordinate hyperplanes of
dimension k—j. We then have the following result given by Davenport [67].
Lemma 3.7 Let s$ be a convex body in Uk and define the functions Aj as in
Section 3.1. Then
N(rf) = Krf) + o(jj a\ (3.5.1)
where /() denotes k-dimensional Lebesgue measure.
Proof We shall write kt() for ^-dimensional Lebesgue measure in the following.
The proof is by induction; the case k = 1 is trivial, and the case k = 2 should also
be evident. We write x for a vector (jc£, ..., xk) e Rk, y for a vector in Uk~ *, and
e = (l,0,...,0)elRk. We let s/(x) be the intersection of s$ with the hyperplane
xe=x, and note that si(x) is a convex body for all x. We let Aj(x) be the sum of
the k— j—\ -dimensional volumes of the projections of $4{x) onto the coordinate
hyperplanes of dimension k—j —I. We let & be the projection of si onto the
hyperplane jte=0, and let #,. be the sum of the volumes of the projections of 28
Proof of Theorem 3.5 83
onto the coordinate hyperplanes of dimension k—j — l. We also let s#(y) be the
intersection of s& with the line jc = xe+y{xeW), and let g(x,y) be the characteristic
function of s& and put
x(y)J° if^> = 0
kKy) (1 otherwise.
We now suppose that (3.5.1) is true in IRfc_1 and try Uk. If & is the intersection
of a body with a hyperplane parallel to the coordinate axes, we write N*(@) for
the number of lattice points in the projection of Si onto the corresponding
coordinate hyperplane. Thus, by our inductive hypothesis,
N*{st(x)) = Xk_ !(.*(*)) + o( X *j(x)
Now
N*(,tf(x))dx =
X g(x,y)dx= £ lastly)),
yeZk 1 yeZk l
while
li{j*(y))=N*{s/(y)) + 0(X(y))
by the 1-dimensional form of the result. Hence
I W0>))= I N*(^(y)) + 0(N(m)
yeZk~l yeZk~l
=N(d) + 0(N{@)).
Again, using the k — 1-dimensional form of the result we have
Also
j=L \j=l J
r fc-i / k \
X Aj{x)dx = 0[ X AX
Thus, assembling all our results leads to (3.5.1) as desired. □
Proof of Theorem 3.5 Clearly we need only consider aef, and we shall once
more appeal to Lemma 1.5. We may also suppose that V(r)>r~2l2 for all r, since
any bodies with smaller volume contribute only 0(1) to (3.1.14) for almost all a.
We may suppose that (1 + €)2 < 5/C, since € is arbitrary. Also, as in the proof of
Theorem 3.3, we can deal with the cases V(n)E(n)^l, V{n)E{n)<\ separately.
84 GCD sums with applications
Now let m,t be a fixed pair of integers and for m^r<t write
fl if arxe@(r) + m, with meA
r [0 otherwise.
We shall want to estimate
I (gr(x)-V(r))) dx. (3.5.2)
We write
@(R,QltQ2) = {reZ.: m^r<t,R<r^R1+€,
Ql+<<2Ah(r)^2Qh{h = \a)},
when Q2>(\og3R)~10'. For £2 ^0°g3i*)~10 the condition on a2(r) becomes
y42(r)^2(log3^)-10. In the following it will be assumed tacitly that R takes on
the values
1 9 9I+€ 9(1 +€)2
while Qi,Q2 take the values 2~^(y = 0,l,...), where
r = 0
By our assumption on V(r) we must have Afo), A2(r)^r~2, and we need only let
Q2 take one value exceeding 2(log3i^)~10. Hence, for each fixed value of R, the
numbers of values taken by Qx and Q2 are «loglog9i? and «logloglog \00R,
respectively. Hence
X (loglog9^)-1-€«l.
R.QuQi
Thus, by the Cauchy-Schwarz inequality, for any real function /, we have
1 /<»Y=( 1 1 /wY
« X (loglog9tf)2+e( X /(r)Y.
Applying this formula to (3.5.2) gives an upper bound
« I I (loglog9R)2+€ ( J gr(x)gs(x) dx- V(r)V{s)
(3.5.3)
Proof of Theorem 3.5 85
By combining Lemmas 3.1 and 3.7 we obtain
gr(x)gs(x)dx
r
= V(r)V(s)+o(mm(v(r) £ W<^Y,K<0 f A{r)fe^\'\\
= V(r)V(s) + o((V(r)V(s))^ £ £(4«4«<^
J + /i\l/2^
(3.5.4)
since Aj(r)^Aj+l(r)^\ fox j^k — \.
The K(r)K(s) term in (3.5.4) cancels out the V(r)V(s) term in (3.5.3), as expected.
The three terms in (3.5.4) in the O notation lead to the three terms in En via
suitable GCD sum estimates. For example, the Qx term in (3.5.4) makes a
contribution to (3.5.3) which is
« Z Q\'2 Z (loglog9r)2+€(>W(s))1/2w(r,5)
R.QnQ.2 r,*6»(Jl,Q1,Q2)
« Z Q\l2 Z (loglog9r)2+< V(r)cxp(^^fDZl
r£,q2 r^ita^) \\og\og(9R1+€)y
(by Theorem 3.2)
5logr
« Z V(r)A\'2-€exJ
m^r<t V
loglog 9r/
The other terms are dealt with similarly using Lemma 3.6 and Gal's result.
To deal with the case V(n)En^\ we note that the argument used to establish
(3.2.6) generalizes to higher dimensions, thus giving
gr(x)gs(x) dx = V(r)V(s) + o(^^
<* \ aras
We then work as in the previous theorems, absorbing the (loglog 9r)2 factor from
Gal's result in the 0((logA(AO)€) term (with A(JV) defined as in the statement of
Theorem 3.5).
The final part of the theorem is obtained by using the conditions
A(«)«exp(-i5gl) and ^2«(log2«)-8^
to simplify the splitting-up argument. Now it is only necessary to subdivide the
range of summation over r. □
86 GCD sums with applications
3.6 PROOF OF THEOREMS 3.6 AND 3.7
We first require a simple lemma.
Lemma 3.8 Let d(ri) be a sequence of positive reals with (5(w)->0 as w-»oo, and
oo
Z <5(") = °o.
Then there is a sequence of distinct positive integers {nk} such that
I */it) = oo (3.6.1)
fc=i
and
d(nk)<k-\
Proof We may assume that S(n) < 1/4 for all n, so that each n belongs to one of
the following sets:
9r={n\ 2-r-3^5(n)<2-r-2}, r=0,l,2,...
Let a = (r: |0r|>2r+1}, and write #r=min(2r+1,|0r|), so that Hr = \9r\ if and
only if r$<%. Now define the sequence nk successively in blocks of length Hr for
r=0,l,2,... More precisely, for H0 + Hx+ •• +Hr_l<k^HQ+Hl+ • +Hr,
we let nk take the Hr values of those ne@r, there being a choice of a subset of Qir
with 2r+1 members whenever re@. This then defines a sequence of distinct
positive integers nk with the property that every value n eQ)r with r ^ is taken by
nk for some value of k. Moreover, each nkeQ)r for some r^O with
A^#0+#1+---+#r^2 + 22 + 23+---2r+1<2r+2,
so that b(nk)<k~l by the definition of @r.
It remains to prove that (3.6.1) holds. First, we have
L<5(«*)=I I *(*)>! L<5(«*)
reSS
so that divergence follows at once if @ is an infinite set. On the other hand, we
also have
where the first sum on the right-hand side is finite if & is a finite set, so that the
second sum has to diverge according to the hypothesis of the lemma. The required
result then follows from our earlier remark on numbers ne3>r with r$@. □
Proof of Theorems 3.6 and 3.7 87
The following result first appears in the literature as a result by Vilchinskii
([268], Theorem 3).
Lemma 3.9 Let if/x («),..., \J/k(n) be k sequences of non-negative reals. Then the
inequalities
Iccjn-mjlKij/jin), («,wJ.) = l, (j = l,...k), (3.6.2)
have infinitely many simultaneous solutions for either almost all
<*(= (<*!, • • •, afc)) 6 Uk or for almost no a e Uk.
Proof Clearly we may restrict our attention to <xe [0, l)k, which we now denote
by <%k (and in general <%j=[Q,l)J). We write a = (x,y) with xe^jef^!. The
proof will be by induction. We first note that the set of a for which the inequalities
have infinitely many solutions is clearly measurable, being the countable
intersection of a countable union of boxes. Indeed, the set is measurable with respect to
the product measure on Uk, and this will be important below, as we implicitly use
Fubini's theorem. Let x(jc, y) be the characteristic function of this set. There are
no problems in swapping orders of integration with this function.
Suppose the theorem has been proved in A: —1 dimensions. The
one-dimensional case is Theorem 2.7(B). By the induction hypothesis, for each x, the value
for x(x,y) is either 0 for almost all ye^k.l9 or 1 for almost a\\yE^k_1. To see
this, replace \J/j(ri)(j^2) by \J/j(ri)d(n), where
fl if \xn—m\<\j/x{n) with (w,«) = l for some weZ
[0 otherwise.
Similarly, by the one-dimensional form of the result, for each y, x(jc,<y) = 0 for
almost all xe%u or x(jc,,y) = l for almost all xe°UV It follows that each of the
integrals
X(x,y)dx,
o
X(x,y)dy
can take only the values 0 or 1.
Now suppose that the measure of the set we are considering is less than 1,
that is,
i r
X(x,y)dydx<\
(and here we have appealed to Fubini's theorem). Since
X(x,y)dy=g(x), say,
is zero or one, there must be a set si <= ^ with 2(jaO>0 and 0(jc) = O for xestf.
88
GCD sums with applications
Hence
g(x)dx=0, and so
s/
X(x,y)dx = 0
.vf
for almost all y, say for all y e@. Thus, for ye 0$,
n
X (x, y) dx ^ 1 - X(srf) < 1.
Since the integral above can only be zero or one, we conclude that
X(x,y)dx = 0 forye@,
that is, for almost all y. Thus
n
X(x,y)dydx = 0
and this completes the proof by induction.
□
Proof of Theorem 3.6 By Lemma 3.9, we need only prove that there are infinitely
many solutions to (3.1.17) for a set of positive measure in °Uk. We do not increase
the number of solutions by supposing that C<l/2, so this will be assumed
henceforth. We write
E(n) = d(n)((p(an)/an)\
r = 0\ am <*m J
and
6m = <ym{\)x... x<9m\K).
IfE(n)-h+0, then A(<fJ++0 so
(00 00 \
n u'- >o
n=1m=n J
and this completes the proof. We may therefore suppose that E(ri) ->0, and so, in
view of Lemma 3.8 and the first Borel-Cantelli lemma, we can assume that
n'1 >E(ri) ^n~2.
By Lemma 2.3 we then need only show that
X X{SmnSn)«[ X E(n)
(3.6.3)
Proof of Theorems 3.6 and 3.7 89
We put
(0m,an)2(log2m)(log2w)
A (m, n) = .
amanil/j(.m)\l/j(n)
Using Lemma 2.12 for eachy, we obtain
X(gmc\£n)«E(m)E(n)
if
max ^4j(w,«) ^ 1, (3.6.4)
j
and
K#m nO« E(m)E(n) (loglog(3«w))fc
if (3.6.4) fails. It follows that
X ^wnO«f Z Wf + S,
where
5= X Z £(m)£(H)^m,H)1/2(loglog(3m«))k.
We now apply Theorem 3.2 to S (with € replaced by e/2). Since log(2m)log(2«) x
(loglog(3w«))k«(m«)€/4, we obtain
By (3.1.16),
EWij/jiny1 «E(nY«n-*
for each y, so that
which establishes (3.6.3) and so completes the proof. □
Proof of Theorem 3.7 Write I(ri) = \J/ (ri)q>(an)/an. As above, we can suppose that
I(n) ->-0. For case (i) we suppose that n~2^I(n) <n~l. We take cr = l/4 in (2.6.11)
and thus wish to bound
Z= Z (^(w)iA(«)log2(m)log2(«))3/4^^^^H;(r,5)1/4. (3.6.5)
1/4 l<m<n<N an am
90 GCD sums with applications
Now, since an is a subsequence of the A:th powers, an=bkn for some sequence of
distinct integers bn. Hence
{{bm,bn)2S
w(r,5)< r-j^-
so we can apply Theorem 3.2 to (3.6.5) with ar replaced by br. Since
\l/(nfl2(cp(an)lan)2<I(nfl2<n-Z12, we have
£« S {\ogn?n-*2 expf-^^_ 1<<L
1/4
Hence
X Wmn<0«(x £(«)Y,
as required to complete the proof.
Now consider cases (ii) and (iii). We take <r = l/2 and so obtain, after an
application of Theorem 3.2, that
X«I^(«)^Y«i. (3.6.6)
1/2 n = l \ an
Hence, in case (ii),
I«I m,
1/2 n=l
which is a suitable estimate. If (iii) holds, we obtain
N / N \2
YJ«NsYJm«( Z m
1/2 n = l \/?=l
infinitely often. In view of Lemma 2.3, this is a suitable estimate. Case (iv) follows
from case (ii), since if we assume I(ri)<n~ *, then
for^CR + 1)-1.
For case (v) we do not need the GCD estimates used previously. We let R(ri)
be the nearest integer below
20c~1«1~<5log2«.
If m<n — R{n), then it follows from the condition an+l/an>\ + cnd~1 that
C"'^2^ «n-.o.
Proof of Theorem 3.8 91
We therefore need only consider
N n N n
X I k{*mn*n)« X /(H)loglog9« X W- (3-6-7>
n=lm=n-R(n) «=1 m = n-£(/?)
If (a) holds, then we obtain the required bound immediately. Indeed, the condition
could be weakened to
q>(an)ij/(n) ^ nd~l » y(Q»(m)
an <<log«loglog9«mt'1 am
On the other hand, if (b) holds, then
m = n-.R(n) W -^W m=l
Since <p(r)»r/(loglog9r) (Theorem 328 of [111]), if a„«expexp(^(log«)_1 x
(loglog9«)_1), it follows that
JR(«)loglog9« q>(am)
—-— « mm .
n — R{n) m*kn-R(n) Clm
This again leads to a satisfactory bound for (3.6.7) and completes the proof. □
3.7 PROOF OF THEOREM 3.8
Clearly the result of Lemma 3.9 holds without the condition (n, mj) = 1 imposed
(just replace the appeal to Theorem 2.7(B) in the proof by Theorem 2.7(A)), so
that we need only prove that there are infinitely many solutions to (3.1.18) for a
set with positive measure. For simplicity we will restrict our discussion to the case
k = 2 (from which the general result could be deduced with more work). We write
Q (r-^)r+!f1W\xM2(^+^)\ 3
r,s=i V an an J \ an an j
(r,s)=l
(where the intervals are interpreted modulo one). By Lemma 2.4,
We write E(ri) = \j/ x(ri)\l/ 2{n). As previously, we may assume that E(n)<n~l. The
coprimeness of r and s in (3.7.1) plays a vital role in the proof.
Now suppose that £nr\£m^0 for some m^n. The measure of overlap is
f\lf2(n) il/2(n)\ ,, ,
i , = <$(m,"), say.
V an "m J
Write, for j = 1,2,
Aj=2m2Lx(\J/j(n)am,il/j(m)an).
^ mm , mm
un um
92 GCD sums with applications
Then Snr\Sm^0 only when there is a solution in rj9Sj to the simultaneous
inequalities
kja»~a»5il<i4i 0*=1>2)> U^fl„, l^^^a„. (3.7.2)
Since (r1,r2) = l=(51,52), there are no solutions to (3.7.2) with
ft a —Si a =0 = t*->q —a s<>.
' l**n "l"™ v '2**/? **m"2*
Now for7 = 1,2 let Jfj be the number of solutions to (3.7.2) with rjan—amSj=0,
and let X3 be the number of solutions with \rjan—amSj\ >0 for bothy = 1 andy = 2.
By the argument used in Section 2.3, we have
JL o ^ ^A i A 2 •
Now the number of solutions to
ra —a s=0, l^r^a , l^-s^-a
is (am,an). Hence (working as in Section 2.3 for one inequality)
Xj^2A3.j(am,an) (y = l,2).
Thus
X Ai(^nO< I ^(m,«)(4^1^2 + 2(flm,fl;?)(^1+^2)).
We have
4 X S(min)A1A2^\6[ £ £(«)),
l^m,n^N \n=l J
which is a suitable estimate. Also
«5(m,«)(am,a„)(^1 + ^2)
^ w(m, ri){E(n)E(m))1i2(<!l'i (™)<Ai ("))1/2 + GM^C*))1'2)-
Thus an application of Theorem 3.9 gives
/ n \2 n
Z n(*mr>*m)«( X £(/!)) + X £(«)(iA1(«) + ^2(«))«€.
l^m./isgN \n=l / n = l
By (3.1.16) and our assumption E(n)<n~*, we have
\j/l(n) + \J/2(n)«n
.-€
This gives a suitable bound for (3.7.3) and so completes the proof by Theorem 2.3.
□
Notes
93
NOTES
The correct maximal order of
V (am>an)
*" (a a V12
Km.nsSN \umun)
remains an interesting open problem. Another problem is to find ways of tackling
the problems discussed, without using such sums. One approach, due to Schmidt,
is considered in the next chapter. Of course, Theorem 3.1 immediately gives a
generalization of our earlier result on blocks of consecutive zeros in the expansion
of almost all a. One can now consider blocks of any integer repeated with a
suitable choice of /?„ and /(«).
The result of Theorem 3.5 can be improved if the convex bodies are 'smooth'.
Of course, it is a well-known fact that the worst bodies in which to estimate lattice
points have straight sides [143]. If each &(r) is a ball, and k^3, then we obtain
the formula (Theorem 2 in [115])
\j/(N) + 0(iJ/ll2(N)(\ogiJ/(N))3l2+€).
One can see sums involving common divisors enter in the related context of
solutions to Diophantine inequalities on manifolds [75]. The results of Chapter 3
can be generalized to several linear forms (see Chapter One of [254]). We give the
case of one linear form below as Theorem 9.1.
4
Schmidt's method
The sequence an. The sequence ctf(ri) where f(x)eZ(x). Application to
continued fractions. Generalizations to higher dimensions.
4.1 STATEMENT OF RESULTS
The results of the previous two chapters have left us in the unsatisfactory position
of knowing that, if il/(n) is non-increasing and
£ *(*) (4.1.1)
diverges, then there are infinitely many solutions for almost all a to
|a«—m\<\j/(n),
but not knowing whether the asymptotic formula for the number of solutions with
n^N holds unless
( £ iKri)\og2nJ (log( £ *(/i)log2#i)Y+<=</ £ <K«)\
This precludes the most interesting cases where \j/(n) = 0(n~1). This problem was
solved by Erdos in 1959 [90] and Schmidt in 1960 [236]. We shall follow Schmidt's
method, which gives a much more precise result as follows.
Theorem 4.1 Suppose that if/(ri) is a non-increasing function with 0<if/(ri)^j and
such that the series (4.1.1) diverges. Then, for almost all a, the number of solutions
S(ol,N) to
\\(xn\\<\J/(n), n^N, (4.1.2)
satisfies
S(oi,N) = 2x¥(N) + 0(x¥(N)1,2(\ogx¥(N))2+€) (4.1.3)
Statement of results 95
for every €>0. Here,
n=l
There are no difficulties involved in generalizing this result to higher
dimensions: we shall give a more general result below as Theorem 4.6. We remark that
Schmidt [236], [240] required a factor
(1,?)'
in the error term of (4.1.3), which we have dispensed with. This factor is 0(1) in
the most interesting cases, so we have not made a significant improvement to
Schmidt's result. It is possible to restrict the integers n in Theorem 4.1 to a given
set and so obtain the following result [117], which we shall see later has some
interesting corollaries (Chapter 6).
Theorem 4.2 Let the hypotheses of Theorem 4.1 be given, and suppose that stf is
an infinite set of positive integers. Write S(s#, a, N) for the number of solutions to
\\<xn\\ <\J/(n), n^N, nest. (4.1.4)
Then, for almost all a,
S(^,oc,N) = 2x¥(N,^) + 0(x¥(N)1/2(\ogx¥(N))2+€) (4.1.5)
for every €>0. Here,
«=i
ties/
Using this result, we obtain an asymptotic formula for the number of solutions
to (4.1.4) when s& is any set with positive lower asymptotic density, in particular,
integers in an arithmetic progression, or the set of square-free numbers. Sadly,
with the result as stated, we just fail to obtain a result when \j/(n)=n~1 and stf is
the set of integers representable as the sum of two squares (though from Chapter 2
we know that there are infinitely many solutions in this case). We leave it as an
exercise for the reader to modify the proof of Theorem 4.1 so that n takes only
values which are sums of two squares both in (4.1.2) and in the definition of *F(iV).
P. Sziisz [260] proved (4.1.5) in the case where s$ is an arithmetic progression
modulo d. However, in place of ^(JVV1 for the main term (which follows from
our Theorem 4.2), he obtained a complicated expression which does simplify to the
correct one. This expression arose through working first with \otn—m\<\J/(ri),
where (m,ri) = \.
Schmidt generalized Theorem 4.1 to several polynomials in an inhomogeneous
situation. There are no great difficulties in extending Theorem 4.2 in the same
manner. Here we give a quick proof of the following result.
96
Schmidt's method
Theorem 4.3 Let \j/(ri) be a non-increasing function with 0 < \j/in) <j and such that
the series (4.1.1) diverges. Suppose thatf(n) is a polynomial with integral coefficients.
Let s4 be an infinite set of positive integers and write £(«£/, a, N) for the number of
solutions to
\\ocf(n)\\<il/(n), n^N, ned. (4.1.6)
Then, for almost all a, we have
S(<tf, a, N)=2V(N, ji) + 0(*F(AO1/2 (log ¥(A0)*+€), (4.1.7)
where B is a constant depending only on f € > 0 is arbitrary, and ^(N, srf) is as
defined in Theorem 4.2.
We can use the ideas used in the proof of Theorems 4.1 and 4.2 to prove the
following result which generalizes the theorems of Sziisz [260] and gives an explicit
error term almost as good as those of (4.1.3) and (4.1.4) (no explicit error term
was given by Sziisz). This result leads immediately to results on the metric theory
of continued fractions.
Theorem 4.4 Let \j/(ri) be a non-increasing function with 0 < if/(ri) ^j and such that
the series (4.1.1) diverges. Suppose that stf is an infinite set of positive integers. Write
S{srf, a, N) for the number of solutions to
\an—m\<\J/(ri), n^N, nejrf, (m,n) = l. (4.1.8)
Then, for every € > 0, for almost all a,
S(jtf,ot,N) = 2 Y \J/(n)— + OC¥(N)ll2(\ogx¥(<N)y+112). (4.1.9)
n=i n
Corollary Let qn(cc) be the denominator of the n-th convergent to the continued
fraction expansion of a. Then, for almost all a and any number B > 7t2/6, we have
qn{0L)<QBn (4.1.10)
for n>n0(oc,B).
Remark By the metric theory of continued fractions, somewhat more is known,
namely
li %2
for almost all a. This was proved independently by Khintchine [158] and Levy
[184] in 1936. See Chapter V of [229].
We note that Theorem 4.4 gives a quantitative version of the Duffin-Schaeffer
theorem for all situations were $0 has positive lower asymptotic density. For
example, we may take sd to be an arithmetic progression or the set of square-free
integers. The Corollary was first proved by Khintchine [153] by a direct method
Proof of Theorems 4.1 and 4.2 97
(and with a larger value for B) and used to establish Theorem 2.2. We now give
some further results on continued fractions which could be proved directly, but
which follow quickly from our previous results. These were first obtained by Borel
[45] (although this paper contains some errors), Bernstein [36], and Khintchine
[153].
Theorem 4.5 Let g(n) be a positive non-decreasing function such that
I 9inyl (4.1.11)
converges. Then, for almost all cc, there are only finitely many solutions to
an(ot)>g(n). (4.1.12)
Also,
N
Xan(a)<<Ar<7(logA0, (4.1.13)
n=l
and
I^^^AWogAO. (4.1.14)
Here we have written ot = [a0,al,a2,...] in a standard notation (Chapter 10 of
[111]). Moreover, if (4.1.11) diverges, then for almost all cc there are infinitely many
solutions to (4.1.12).
It can be seen from this result that the simple aspects of the metric theory of
continued fractions are in a very satisfactory state. On average, we find that
an«g(\ogn), where g is any positive non-decreasing function such that (4.1.11)
converges, although infinitely often an»g(ri) for any g such that (4.1.11) diverges.
By (4.1.10) we have
(N \1/N
Ylan) ^ exp(7T2/6).
n=l /
In fact Khintchine [157] proved that
N \l/N oo / j \^
as JV->-oo. The estimate (4.1.14) was applied by Khintchine to estimate the
discrepancy of the sequence not. We shall do likewise in the next chapter.
4.2 PROOF OF THEOREMS 4.1 AND 4.2
We begin by outlining the basic ideas involved. The reader should now be very
familiar with overlap estimates. In Chapter 2 we obtained estimates of a suitable
98
Schmidt's method
size because the centres of intervals did not coincide. This was a beneficial
consequence of demanding (w,«) = l when considering |aw—n\ <\J/(ri). The
method of Cassels also depended on ensuring that the centres did not coincide,
by removing all the intervals for which this happened. In the previous chapter we
obtained the expected value \J/(ri)\J/(m) for the overlap estimate with an error which
again corresponds to the number of times the centres of intervals coincide.
Schmidt's more subtle approach entails removing some, but not too many, of the
coincident fractions. To precise, we consider
\ocn-m\^\J/(n), (/w,/i)<*F2(/i) + l, n^N (4.2.1)
in place of ||an|K^(/i). We shall demonstrate that the restriction (m,ri)^
*F2(h) +1( = T(n), say) reduces the contribution from the intervals whose centres
coincide to manageable proportions. On the other hand, this condition does not
reduce the number of solutions to (4.2.1) by a significant amount for almost all a.
We write
(m, n)<r(n)
Then
n
where
m=l
(m,n)^T(n)
We also write
<p(k,ri)= X *>
m=l
(m, n)^k
d\n
A(r,s)= III,
K=lV=l
us — vr = 0
(u.r)^r(r)
(i\s)<r(s)
L(n) = log 3T(/i), L2(n) = log 2L(n).
We now state some preliminary results.
Proof of Theorems 4.1 and 4.2 99
Lemma 4.1 We have, uniformly in M, N and k, that
£ ^^=^_M+1 + of^ + log4 (4.2.2)
Proof Since <p(k, n) ^n, we need only obtain a lower bound of the form given on
the right-hand side of (4.2.2). We have
q>(k,ri)^n-Y, £ 1.
d\n m = 0(modJ)
d>k m^n
Thus
n = M n n = M \ n d>ku
d\n
N i N
-M-N+l- I - I 1
d = k+lu n = M
n = 0(modd)
*r *, 1 V- N~M J £ l
=n-m+\- £ —n-+o[ X -j
d = fc+l " \d=k+lu/
'N—M
=N-M+\ + 0[—-— + log AM,
as required. We remark that our proof is simpler than Schmidt's original, and the
result stated is sharper. □
Lemma 4.2 We have, uniformly in k, M and N, that
I^^ = (l + 0(*-'))L-K«) (4.2.3)
n = M n n = M
+ o((logM)iKM)+ I —}
Proof By partial summation we have
n = M n n = M m = M m m = M m
N_1 / (n-M N
= I (^(/i)-^(n + l))(/i-Jlf+l + 0(—t—+log/i
'N—M
+ \J/(N)\N-M+l + 0[-—— + logJV
100 Schmidt's method
(using (4.2.2))
= (l + 0(/T1)) X Hn) + 0[ J]'(*(/i)-W/i + lWlog/i + ^Wlogiv),
which quickly leads to (4.2.3) upon reversing the partial summation in the error
term. □
Lemma 4.3 We have
N \l/(n)<!>(ri) N
Z *K) K> = Z <K") + 0(L(N)L2(N)). (4.2.4)
„=i n n=i
Proof Since
Z < Z <K")>
n-1 * «-1
we need only concentrate on obtaining a lower bound of the right form. Now let
Mo = 0, Mj=22J fory>l. Let /* be the smallest integer for which Mh>*¥(N). We
note that h«L2(N). We then have
£ »(/i)fl(/i) ^ *"x " (A^)y(Mr2 + l^)
n = M W r = 0n = l H
Mr<<F(n)<Mr+1
N / N ^/^\ fi— 1 / i N
= Z *(n) + O Z — + Z T72TT I *&> + (log K>(Kr)
n=l \n=l n r = 0\Mr+L n = l
(4.2.5)
after an application of (4.2.3). Here Vr is the smallest integer such that *F(Kr) ^Mr.
We consider the error terms in (4.2.5) as follows. We have
Also,
Z(iogFr),KKr)«Z<K^)Z-
r = 0 r = 0 n=lW
« Z " Z Wr)« Z h>
n=in r=l n=l n
M.>n
Proof of Theorems 4.1 and 4.2 101
since \j/(ri) is non-increasing. Because h«L2(ri), it therefore remains to show that
£^W), (4.2.6)
•1=1 n
and (4.2.4) will then follow. By Holder's inequality, for any €>0 we have
€ 1
N \Tnr: / N
S*T«l I«»)1+'"T '(I""1
„=1 n \n=l / \n=l
, ,l+€ / w \l+€
-1 \ / *-• -l-€
Also, for e<l,
N
I"-1"€<2
n=l
We choose € = L(N)~1, and so obtain
00 d* 8
1/2 x fc
■ -1 W
^ W(N)L{NrlL(N)« L(N),
which establishes (4.2.6) and so completes the proof. □
Lemma 4.4 For all n>m^\ we have
Mmn*m)^4<KmMn) + 2^M(w>w). (4.2.7)
n
Proof Let X{Smr\Sn) = B1 + B2, where 5X is the contribution to the intersection
from intervals whose centres do not coincide, and B2 is the contribution from
intervals whose centres do coincide. Thus
B^W0j< " l=2iKn)A(n,m)
2^ n „ti „ti n
um — vn = 0
(u,n)^r(n)
(m,v)s£r(m)
By the method we used in Section 2.3, we have Bl^\J/(n)\J/(m). We now require
a more subtle argument to show that B1^4\l/(n)\lf(m). The problem with our
earlier approach is that it supposed that the measure of the intersection was always
as great as possible. We now show that the individual intervals making up the
intersection have differing overlaps in the expected manner.
Suppose that two intervals overlap, say
'u — if/(ri) u + \J/(n)\ (v — \l/(m) v + \j/(mfx A
, Jnl , )^0.
n n I \ m m
102 Schmidt's method
Then there is some h, with \h\^n\J/(m) + m\J/(ri), and
\nv—um\=h. (4.2.8)
The length of the overlap is, by a simple calculation,
mm , 1 = E(m, n, h), say.
\ n n m mnj
Here we have noted that \J/(ri) ^ \J/(m), since n > m, and \j/{n) is non-increasing. Since
we are interested in obtaining an upper bound, we can remove the conditions
(u,ri)^r(ri), (w,u)^r(m), and thus the number of solutions to (4.2.8) becomes
(m,ri). Hence, upon writing y = n\J/(m) + m\J/(ri), we have
#i^ X (m,n)E(m,n,h)= £ (m,n)E(m,n,h(m,ri))
0<|/»|<y 0<|/»1< y
(m,n)\h (myn)
^2
=2
/*y/(»n,n)
(m,n)E(m,n,h(m,n))dh (4.2.9)
o
<*y
E(m, n, x) dx
o
(by a simple change of variable)
=4\J/(m)\l/(ri)
by explicit calculation. This completes the proof of (4.2.7). □
We remark that we obtained (4.2.9) from the previous line, since E(m, n, h(m, n))
is an even function of h which does not increase as \h\ increases. Clearly, for any
function f(h) with this property,
Z f(h)^2
0<\h\^A
CA
flh)dh.
o
We remark that the expected value for X(gmn£n) is actually
O(w) <&(n)
4\J/(m)il/(n)-
m n
but, in view of Lemma 4.3, this is sufficiently near to 4\J/(m)\J/(ri) not to cause any
serious difficulties. Since A(m,ri)^:(m,ri) Lemma 4.4 provides an alternative proof
of (3.2.4) in the homogeneous case. Now we establish a more useful bound for the
Proof of Theorems 4.1 and 4.2 103
present context as follows. We start with
£ A(m,n)^ X 1.
m=l l^m,r,s^n
r/n = s/m
(r,/?)ssr(n)
Now, corresponding to each r, there are coprime integers a, b satisfying rln=alb.
Here b has to be a divisor of n, and if we fix such a value b and allow r to vary,
with l^r^n, then a has to satisfy an/b = r^n, that is, a^6. In other words, each
divisor b of n occurs precisely b times for l^r^n. Moreover, the number of
solutions to
a s
- = —, \^s,m^n
b m
is nib, so that
£ A(m,n)^ X jb = nd*(n).
m=l b\n °
n/b^r(n)
We therefore obtain
£ Wmn^n)^4ij/(n) £ iA(m) + 2rf*(H)iKH). (4.2.10)
m=u m=u
Proof of Theorem 4.1 We write S*(a, w, i>) for the number of solutions to
\ocn—m\<\J/(ri), (w,«)^T(«), u<n^v.
Clearly
5*(a,M,u)da = 2 £ \J/(n)-^-, (4.2.11)
Jo u<n^w W
and the number of solutions to (4.1.2) is
S(oL,n)^S*(oL,0,N) (4.2.12)
for every N. We therefore have
f1 JL ( OfoV
(5(a,i;)-5*(a,0,z;))da = 2X *(*) 1
Jo n=i V *
= 0(LGOL2(i;)) (4.2.13)
by Lemma 4.3. It follows from (4.2.12) and (4.2.13) that, for almost all a,
S(a, N)=5*(a, 0, TV) + 0(L\N)\ (4.2.14)
as we now justify. Write Giri) = L(n)/L2(ri). By (4.2.13) and (4.2.12), the measure
of the set of a for which
S(a, AT) - 5*(a, 0, TV) > L2(iV)
104 Schmidt's method
is «G(N)~l. Let i>. be the smallest integer n for which L(ri)^2J. Then clearly
oo
j = 0
converges. Hence, for almost all a,
S(a,i^-S*(a,0,^<L2(t;,.)
for ally >y(a). Because S(ol, N)—S*(<x, 0, N) is a non-decreasing function of N, and
L\N)^4L2(Vj) for Vj.^N^Vj, the formula (4.2.14) follows.
To prove our theorem we therefore need only show that
5*(a, 0, AT) = *¥(N) + OC¥ll2(N)L2+€(N)) (4.2.15)
for almost all a. Write
We shall apply Lemma 1.5 in a standard fashion. From (4.2.11) and (4.2.10) we
have
(s*(a,K,t>)-2*F(K,i?)j da
^4 X d*(/#(/i)+4¥(i*,t;) X iA(«)(l- —
« X ^*(«)^(«) + ^(n)I<n)L2(n) + ^l-^W(/i)\ (4.2.16)
To obtain (4.2.16) we noted that
^ X *(n)Un)L2{n)+ £ ^(r)( 1 - ^W)
by Lemma 4.3.
We can use (4.2.16) in applying Lemma 1.5 and so obtain that, for almost all a,
S*(a, 0, N) = 2¥(N) + 0(¥}/2(log V1(iVr))(3/2)+W2)), (4.2.17)
Proof of Theorem 4.3 105
where
N N
+ X>("/l-^W) (4-2.18)
N
by applying Lemma 4.3 to the final sum in (4.2.18). We have
n = 1 n = 1 d|n
<*<r(n)
= I I <K"K I —t-
d^HN) n=l d^r(N) u
n = 0(mod d)
(since ^(/i) is non-increasing)
«*¥{N)L{N).
Hence yVl{N)«yV{N)L{N)L2{N). Thus (4.2.15) follows from (4.2.17) and the proof
of Theorem 4.1 is complete. □
Proof of Theorem 4.2 Clearly we may restrict the appropriate variables to s4 at
each stage of the proof. If S{s4, a, N), S*(jrf, a, u, v), *F(^, u, v) replace S(a, N),
5*(a, u, v), *F(w, v) respectively in the above, then we obtain
S(j*, a, N) = S*(s/9 a, 0, N) + 0(L\N))
for almost all a. Also we obtain
•i
(S*(j* , a, w, t?) - 2¥(^, m, u))2 da
o
^4 X J*(n)^(n)+ 4^(^,11,!;/ X <K*)(l-—)
ties/ nes/
Since all the terms are non-negative, we can omit the condition n e s/ and complete
the proof as above. □
4.3 PROOF OF THEOREM 4.3
The result follows immediately from Theorem 4.2 (by redefining s&) if/ is linear.
As in Section 4.2, the restriction of n to stf is straightforward, so we shall
henceforth suppose that s4 = Z +. By ignoring the first few values of/ we may assume
106
Schmidt's method
without loss of generality that / is positive and strictly increasing for «>1. If
f(x) = g(x)\ where g(x) is a polynomial, for some t ^2, then the result follows from
Theorem 3.1 with a little work. It turns out to be the case, rather perversely, that
in the method we employ here it is the repeated factors of the polynomial which
cause some difficulties. The following is our analogue of Lemma 4.1.
Lemma 4.5 Let f(n) be a positive strictly increasing polynomial for n^l. Then we
have, uniformly in M, N, and k, that
i ^!»^-M+l + o((JV-M:1)10gBfc + log«iv), (4.3.1)
where B is a constant depending only on f
Proof Working as in Lemma 4.1, we have
/(") itfU)d
d>k
We therefore need to bound
AN) 1 N
d = k+lu n = M
f(n) = 0(modd)
We note that it is possible to establish an asymptotic formula for
N
Z «f(n))
n=l
for polynomials of degree 2, and one can obtain upper and lower bounds of the
correct order of magnitude for general degree [89], but this does not lead directly
to our result.
Now, given/, there is a polynomial q(x)eZ[x] such that: (i) q has no repeated
factors; (ii) f{x)\q{x)1 for some teZ+. Hence /(«) = 0 (modd)=>q(nY = 0 (modd).
Now write r(g) for the number of solutions to q(n) = 0 (mod#) with l^n^g. Since
q has no repeated factors, we have r(g)«-cA(g\ where t is the usual divisor
function, A is a positive constant, and the implied constant depends only on q
(indeed, on the discriminant of q). Now any integer d can be written in the form
0102 03---0!*
where: (i) if t ^ 3 we have (#,-, gh) = 1 if 1 ^y <h < t; and (ii) each g} is square-free for
j<t. If d is written as above, put
d'=glg2-..gv
Proof of Theorem 4.3 107
We then have q\x) = 0 (modd)^>q(x) = 0 (modd'). Hence
AN) i N
«I I »
d = k+lu n = M
q(n) = 0(mod d')
*\(N-M+\ „
where * indicates the summation conditions
Now i
Since
K<*>
= E[ r^j) and
so
*
I
9}
1 r(d')
d~lT
Cs£n<2C "
«fhp
gj j=iyj
(logC*)^"1
this gives
for some B = B(A,t). Also
* 1 r(d') \ogBfc
55-7"*— •
X*^«(logiV)-<"
0;
This establishes (4.3.1) as required. D
It follows from Lemma 4.5, working as in the previous section with
0(H) = <p(r(H),/(H)),
that
I ^fv^ = I *(») + 0(L2(N)L(N)B). (4.3.2)
To obtain (4.3.2) we estimated
N
X /T^a+iog/i)'-1
108 Schmidt's method
with€ = L(A0-1 by
X «-1-€(l + log«)B_1«
(\og3xy
—it+t—d*
1/2 ■*
«
fr<N)(log3x)^
vl+€
1/2 ■*
d;c +
r°° (log 3 s)*-1
vl+€
dx. (4.3.3)
The first integral on the right-hand side of (4.3.3) is clearly ^4L(N)B, while the
second, after the change of variables t=\og3x/L(N), becomes
roo
Q-'^-^dt L{N)B«L{N)B.
i
The only other result we need to complete the proof as in the previous section is
contained in the following lemma.
Lemma 4.6 Write
d*(n) = X 1.
d\f(n)
Then
N
X ^/*(«)iA(")«^WBvF(iv"), (4.3.4)
«=i
for some B depending only on f
Proof We first note that by the argument used in Lemma 4.5 we have, for any
H rid)
£_y«(log#)* (4.3.5)
d=i «
for some B, with r(d) as defined in that lemma. Also, for any H, M^l,
H min(d, M) min(M,H)
I I l<< Z Z l«min(M,#)(log(min(M,i/))B. (4.3.6)
d=l «=1 n=l d\f(n)
/(;?) = O(modd)
Now, taking the upper limit of summation as M to allow a partial summation
later, we have
M T(N) M
X </*(*)< Z Z i
/(n) = 0(modd)
T(N) min(M,d) T(N) M
= Z Z i+I Ii. (4-3-7)
/(n) s 0(mod J) /(n) s 0(mod J)
Proof of Theorem 4.4 109
where the inner sum in the final term of (4.3.7) is taken to be zero, by the usual
convention, when M^d. The first term is «M(logr(JV))B by (4.3.6) and the
second is
<rf^«M(.ogrW)«
by (4.3.5). Hence
<-i d
M
X d*(n)«M(}ogr(N))B,
which gives (4.3.4) by partial summation. □
The reader should have no difficulty in completing the proof with
f.in) (m-\\/{n) m + \J/(ny
\= u -
m=l V
We then have
fin) f(n)
1//BN 20(«)
(mod l).
and, for n>m,
An) '
K*mn*H) ^ 4il/(m)il/(n) + 2d*(n)^(n). D
4.4 PROOF OF THEOREM 4.4
As in the previous sections, we shall omit the restriction nejrf, and write S(jrf, N)
for S(Z + ,<x,N). Now write T(d,<x,N) for the number of solutions to
\<xn—m\<il/(n), d\(m,n), (m,n)<T(N), n^N,
where T(ri) = *F4(h) +1 here. We then have
r(N)
5(0, N) = X Kd)T(d, a, N) (4.4.1)
d=l
(/z() here denoting the Mobius function). Now write A(ri) = x¥2(ri) + \. It will be
straightforward to demonstrate that
UN)
Z T(d,oc,N) (4.4.2)
d = A(n)+l
is quite small for almost all a (in fact of size 0(L(N)3)). We then need to generalize
Lemma 1.5 and use the working of Section 4.2 to obtain the expected asymptotic
formula for T(d,<x,N) for almost all a for all d^A(N) simultaneously. This
roundabout method seems to be more efficient than trying to prove the result
directly.
110 Schmidt's method
As in the proof of Lemma 1.5, Let n1, n2,... be defined by
rij=max{«: *F(«) < j).
As in the proof of that lemma, we need only establish the formula (4.1.9) when
N=rij. We have
P™ ma o V *(n) <*>(T(N)/d,n/d)
JO 11=1 ^ TUu
n = 0(modJ)
Thus
ri UN) 2 N
X 7Xrf,«,JV)da<——TIW») I 1
0 d = A(n)+l ^iV^-l-l n=1 d<T(n)
TO,™
Hence, if ai is the set for which
X T{d,a,nj)>Unj)3,
«J = A(n,)+l
we have
00 00
X A((7J)« Z T
1
2„->
,r2 J j=2yiog7
which converges. Hence, by Lemma 1.2, almost all a belong to only finitely many
ajy so that
£ W,a,«i)«L3(«J.) (4.4.3)
d = A(n})+l
for almost all a.
As in the proof of Lemma 1.5, we express j in binary scale as
j= I b{j,v)2\
0sgt;sSlog2j
and write
@(j) = {(i,s):i = X Kj,v)r-; b(j,s) = l O^s^r},
v=s+l
where r=r(j) = [\og2J]- We write
d) _ f 1 if \xnld -m\< ij/(n)/d with (w, /i) < T(N)/d
"n (*)_1o otherwise,
Proof of Theorem 4.4 111
Fd(Ux)= X (fld\x)
n = u0+l
n = 0(modd)
where
and
ut = ut(i, s) = max{« > 0: ¥(«) < (i + 02s},
Gd(r,x) = X Fj(i,s,x).
0<ssgr, i<2r_J
We then have, for every d and every jc,
"J "J \l/(ri)
I^)=I^T + Z Fd(Ux). (4.4.4)
«=1 n=l " (i,s)e#(j)
Working as in Section 4.2 (compare (4.2.16)), we have
f1 "' /<
Fj(i,j,a)da« X I
10 /? = «„+! \
J, fd*(n)ij/(n) + ij/(n)L(N)L2(n) + / 0>(w)\ ¥(n)N
0
n = 0(mod J)
<i <i V n
Here 0(«) = (p(A(«7)5«). Thus, upon writing v=n2r, we have
A(v) ri v
X </ Gd(r,a)da«r£ (d*(n)2iJ/(n) + d*(n)L(n)L2(n)iJ/(n))
'o «=i
d=i
«r^(v)L(v);J«r42r.
Hence, if we write or for the set of a for which
f^C?d(r,a)>r5+€r,
we have
00 00
Z%r)<I''-1-'<00.
r=l r=l
Thus, for almost all a,
£rfGd(r,a)<r5+e2', (4.4.5)
d=i
for r>r(a).
112 Schmidt's method
Now suppose that a belongs to the set for which (4.4.5) holds and r=[\og2J] +
l>r(a). Then, by the Cauchy-Schwarz inequality, we have
Mnj) /Mij) \l/2/A(v) y/2
d=l (i,s)e3t(j) \d=l (i,s)e^(j) / \d=l ,
^(iogA(«,.))i/2i^o")r/2(r5+€)i/2
<<¥(w,.)1/2L(w,.)7/2+€.
Thus, using (4.4.1), (4.4.3) and (4.4.5) we obtain
S(«> nJ> = Z ^T- I ^(w) + °(L(wi>3) + Omnj)ll2W"2 +€). (4.4.6)
nsO(modd)
We have
n.
1-3- Z *(*)= 2>(*)i-j— Z ~r Z *(*)
d=l u n=l n=l J|n u d>A(n,) " n=l
n = 0(modJ) n = 0(modJ)
= iw«)^+oi i d-2«p(«;)
since \J/(n) is non-increasing,
n = l W \d = Mn})
=i^)^+o(i).
This together with (4.4.6) establishes the required formula for n=nj and so
completes the proof. □
4.5 THE METRIC THEORY OF CONTINUED FRACTIONS
We first state some fundamental properties of continued fractions (see [229], or
Chapter 10 of [111]). We have, for any irrational a,
1
<
qn
< ; (4.5.1)
QnQn+l
|a—alb\<{2b2)~l, (a,b) = \^>a=pn, b=qn for some n; (4.5.3)
a=pn, b=q„=>\(x-a/b\<b~2. (4.5.4)
Now, from Theorem 4.4 with \J/(n) = l/(2«), we find that for almost all a the
number of convergents with denominators qn^M is at least 7r2/61ogM+
0((logM)1/2+€). Hence
n>^\o%qm + (MogqH)1,2+% (4.5.5)
which gives (4.1.10) and so proves the corollary to Theorem 4.4.
The metric theory of continued fractions
113
From (4.5.2) we deduce (since af^-X fory'^1) that
Thus (4.1.14) follows from (4.1.13). Also, using (4.5.6) with (4.5.1) we obtain
1
2(0„+i + D<7n2
<
Qn
<
1
<**+!&
(4.5.7)
If an>g(n), then, by (4.5.7) and (4.5.5) (used to compare n and qn), we have, except
for a set of a of measure zero,
l#na-All<(?„0(log#„)r1-
Since the convergence of (4.1.11) implies the convergence of
00 J
y —-—
Bfi«0(log/i)'
we obtain (4.1.12) from Theorem 2.2.
Now, in the same way that we obtained (4.5.2) we can use (4.5.4) to show that
N^\ogqN + 0((\ogqN)ll2+()
for almost all a. Write/(«) = 2^(2« + l) + l. Then, if (4.1.11) diverges, so do
tfinr1 and £ (/i/flog*))"1.
(4.5.8)
n=l
n=l
It follows by Theorem 4.5, Corollary 1 that, for almost all a, there are infinitely
many convergents pN/qN with
a —
Pn
Qn
<
1
qlf^ogqN)'
Thus, by (4.5.7),
Hence
2(an+1 + l)>/(log^).
an+l>g(\ + 2\ogqn)^g{n + \)
by (4.5.8). This establishes the divergence case for (4.1.12).
To prove (4.1.13) we require the following simple lemma.
Lemma 4.7 // g(x) is a positive non-decreasing function for which (4.1.11)
converges, then, except for finitely many n, g(ri)>n.
114
Schmidt's method
Proof Suppose that g(ri)^n for infinitely many n. Then there must be an infinite
sequence ni with ni > 2h,-_ x and g(nj) ^ rij. Thus
00
00
1
00
Zri»)_I>Z Z -=Zi=°°
n=l
j=l (l^n^iKn^j J=l
This contradiction proves the lemma as required.
□
We may henceforth suppose that g(ri)>n, and clearly we may also assume
g(ri)<n2. Now let S(A, a,N) denote the number of solutions to
2N
\(xn—m\<(nA) \ (m,n)^ 1 + (N/A)2, n<e
For almost all a, by (4.1.12) if (4.1.11) converges, we have aN«Ng(\ogN). Thus,
using (4.5.7) and (4.5.4) we get
N
N
-1
«£^S04,a,A0,
where A takes on the values 2J, y' = 0,1,2, ...,r. Here we have written
r=l + [log2^(logA^)]. Using the methods of section (4.2) we have
"i / 4jv\2 N
S(A,ol,N)-—) da«-max(l,log(JV7,4)).
(4.5.9)
We also note that
X All2Nm max(l, logC/VM))« Ngm{\og N).
We have
(4.5.10)
[ (Z /^>a> #) - ^))2 da ^ (Z a( ['(siA, a, iV) - ^Y daY''V
«JV"20(logAO
by (4.5.9) and (4.5.10). Hence the set of a for which
4/V
ZA\S(A,<x,N)- a
> Ng(\ogN)
has measure «g(\ogN)) 1. We let N take the values 2r. By the convergence of
(4.1.11) we deduce that, for almost all a and N of the form 2r, we have
ZAlSiA^N)-—) « Ng(\ogN),
A generalization to higher dimensions 115
and so
YAS(A,(x,N)« Ng(\ogN) + N\ogN« Ng(\ogN)
A
(at this point we needed g(ri)»ri). Since
A
is a non-decreasing function of N and Ng(\ogN)«(N/2)g(\og(N/2)), this
completes the proof. □
4.6 A GENERALIZATION TO HIGHER DIMENSIONS
Gallagher [105] described a body ^<=[0,l)k as having 'property &* if
xe& =>ye@ whenever O^yj^Xj.
He then proved that if 0&n is a sequence of such bodies with A(&n) = \J/(n), with
\j/(n) non-increasing and (4.1.1) divergent, then there are infinitely many solutions
to
n<x+me@n, meZk
for almost all a e U\ and with all components of m coprime to n. One interesting
corollary to this result (coupled with the easy convergence case) is that there are
infinitely many solutions to
H(l0gH)k + *riK"ll<l
J'=l
for almost all ae Uk if (5^0, but only finitely many if d>0. To see this, take
^n = {(ai, ...,afc)e[0,l)k: (Ka1...ak^H"1(log«)"k"<J}.
Of course, the result for boxes with one corner fixed at the origin in [0,l)k also
follows, and it is not difficult to deduce a result with boxes centred on the origin
as in Theorem 3.8. Gallagher's method combines naturally with Schmidt's method
to give the following result [277].
Theorem 4.6 Let &n be a sequence of bodies with property 0, such that
X{@n) = \j/{ri), where if/(ri) is a non-increasing function such that (4.1.1) diverges. Then
the number of solutions to
n<z+me@n, meZk, l^n^N
is, for almost all <xeUk,
x¥(N) + 0(x¥ll2(N)(\ogy¥(N))2+e), (4.6.1)
116
Schmidt's method
with
¥(AD=2>(")-
n=l
Remark As noted in Chapter 3, the above is stronger than Theorem 3.8, in that it
gives a quantitative conclusion, and the only restriction on shape is 'property &\ It
is weaker in that it requires ij/(ri) to be monotonic. The reader will easily verify that
it is possible to change the proof to give infinitely many solutions with m having all
its components coprime to n. Also, there are no difficulties in replacing n with a
polynomial and restricting n to a given set as in Theorems 4.2 and 4.3.
We define 0(«), A(m, n), and T(n) as in Section 4.2, and put
<f„ = {ae[0,l)k: na+mel,, (mpn)<T(n)}.
Then, since xk^kx—(k — 1) for jc^O, we have
Thus, by Lemma 4.3,
Z W.) = I *(*) + 0(L(N)L2{N)). (4.6.2)
For the overlap estimate we need first a lemma which gives one useful
characteristic of sets with property ^.
Lemma 4.8 If <$ and Q) have property 0>, then X{$r\{Q) + t)) is a non-increasing
function of \ t}\ (1 ^ j ^ k), both for tj ^ 0 and for tj ^ 0, when the other components
of t are held fixed.
Proof The case k = 1 is evident, since # and Q> here would be intervals of the
form [0,x) or [0,x] with 0 < jc^I. The general case reduces to this case since, if
we write X\> Xi f°r tne respective characteristic functions of # and @, then
A(#n(0 + *)) =
Xi(x)l2{x-t)dx,
and, for fixed values of xn(n^j), xm(x) (w = l,2) is the characteristic function of
an interval of the form [0,y] or [0,y).
The following lemma, which is based on Lemma 2 in [105], gives the required
generalization of Lemma 4.4. As we have seen in Chapter 3, the most difficult
case is k = \. Since results for this case can always be deduced from higher-
dimensional results, we expect the biggest contribution to the error in the overlap
estimates to come from a projection onto a one-dimensional hyperplane (a line!).
Because we can deal with the one-dimensional case for monotonic ij/(n), we obtain
the required result. Theorem 4.6 follows immediately from Lemma 4.9 and (4.6.2),
A generalization to higher dimensions
117
using the method of proof of Theorem 4.1. We recall that
u=lv=l
us — vr = 0
(u,rKr»
(v,s)^T(s)
Lemma 4.9 For all n>m^\, we have
Proof At first we must establish that
n
(4.6.3)
* \l/(ri)
WwnO <*(")*(")+I^r>
n
(4.6.4)
where * indicates summation over ul9..., uk,rl9..., rk with
u- r ■
-1 _ _i
<— for all/',
m
(4.6.5)
and
We have
mUj=nrj for at least one j,
0^Uj<n, 0^rj<mi (Wj,«)^T(«), (rj,m)^r(m).
,mn,m.(J(is±Ln^+'"
(4.6.6)
where the union is over all vectors r,ueZk such that (4.6.6) holds. Since n>m and
\J/ is non-increasing, the contribution from r,u satisfying (4.6.5) and (4.6.6) is no
more than the second term on the right-hand side of (4.6.4). It remains to show
that the measure of the contribution from r,u satisfying (4.6.6) but not (4.6.5), say
A(m,«), satisfies
A(m, n) ^ \j/(m)il/(n). (4.6.7)
We note that
0<
n m
<—for all /
m
in this case. Now, if r'n — u'm=rn—um, then u' = u mod(n/(m,n)). Thus the
number of pairs r, u for which rn—um takes on any given value is at most (m, n)k.
Thus
Km.nj^^K/YipnP + j
(m, nf
mn
= Yj Ks, m, n), say.
118
Schmidt's method
Here the summation is over those seZk with no coordinate zero. To each given
s we associate the unit cube
J(s) = {a:Sj^(Xj<Sj +1 if Sj<0,Sj—\<a,- if Sj>0}.
By Lemma 4.8 we have
2(s,m,«)^ /l(a,w,«)da.
JS(s)
Since the J{s) are disjoint, this gives
X(m, n) ^ (m, ri)h
= (mn)k
/L(a,m,«)da
w
x(^J^+a]]da
with a simple change of variables
= ^(m)^(«),
since, for any two measurable bodies <f, &9 we have
U0 n(3? + a)) da = K$)K&)-
w
Thus (4.6.4) is established.
Now, for fixed m, n the number of solutions to
-
«
_0
w
1
< —
m
is at most In for each y (for each value of u} there are at most two values of r,).
Hence
«
«
where ** indicates summation over um = rn with (u,n)^T(n), (r,m)^T(m).
In view of the definition of A(m,«), this completes the proof of (4.6.3). □
NOTES
There may be further applications of Schmidt's method to be discovered, where
the conditions on \j/ are relaxed somewhat, or n is restricted to certain classes of
sequence (other than the polynomial ones discussed here). The reader has now
encountered the major ways currently known for tackling overlap estimates. Each
has its own strengths and weaknesses. Can the reader find a new way of producing
these estimates, or derive a method which sidesteps them?
Notes
119
There may be other natural classes of bodies for which a result like Lemma 4.9
holds true. Can the reader find any? Theorem 4.4 with the quoted good error term
has not previously appeared.
We proved Theorem 4.5 in this context chiefly because we need the result in
the next chapter when we discuss the distribution of the sequence «a(mod 1). The
natural measure on [0,1) to use when discussing continued fractions is, of course,
M*)=; l
dx.
,1+x
log 2
If we write T for the continued-fraction transformation
fO if * = 0
{{1/*} ifjc#0,
then, for any measurable subset of [0,1), n(S)=n(T£) and S = Ti if and only if
\i(£) is zero or one. In other words, T is ergodic. Also, it describes a chaotic
dynamical system with periodic orbits corresponding to quadratic irrationals. A
fuller discussion of the metric theory of continued fractions may be found in [159]
and [229] (see also the interesting results of Philipp [213], which have a similar
flavour to some of our theorems). However, it is of some interest to see that some
of the results on continued fractions can be derived as here, rather than by more
direct methods.
Uniform distribution
Definitions. The sequence <xbn and normal numbers. Uniform
distribution and Riemann integration. Koksma's inequality. Fourier analysis. The
Erdos-Turdn theorem and the Weyl criterion. The sequence n<x. Very
slowly growing sequences. Metrical theory. General sequences anx. The
Davenport-Erdos-LeVeque criterion. Discrepancy estimates: upper and
lower bounds for an<x and fn(cc). Upper and lower bounds for noc.
Marstrand's theorem. Extensions to higher dimensions. The Erdos-
Turan-Koksma inequality. The sequence net. Discrepancy using tilted
boxes.
5.1 DEFINITIONS AND ELEMENTARY PROPERTIES
Let (xn) («= 1,2,...) be a sequence of real numbers. We then define the
discrepancy of (xn) modulo one, DN(xn), by
DN(xn) = sup
./c[0,l)
X \-Nl(J)
n=l
(5.1.1)
where J denotes an interval (open, closed, or half-open). In (5.1.1) we note that
DN(xn) depends on the sequence (xn), not just on an individual term xn. A similar
convention will hold for the definition of weighted discrepancy below. We shall
call a sequence (jc„) uniformly distributed (modulo one) if
limM^=0. (5.1.2)
N-oo N
The following theorem shows the equivalence of our definition with another
common definition.
Theorem 5.1 The sequence (xn) is uniformly distributed (modulo one) if and only if
lim4 I \ = P-ol (5.1.3)
N^oo^ n=l
for every pair a, (3 with 0 ^ a < j5 ^ 1.
Definitions and elementary properties
121
Proof The 'only if part of the theorem is clear, since (5.1.2) is stronger than
(5.1.3) because it is uniform in intervals J'. Now suppose that (5.1.3) holds. For
a given €>0, put k=4[e~1]. Then there is an N(e) such that, for all N>N(e),
1 N
iV n=l
€
<2
for every pair a,/? of the form a/k,b/k with O^a^b^k. For each interval
J a [0,1), we can pick a, b such that
./£
~a-\ b + l
with
and
Thus
[ak-^bk-^Jf ifl(J)>2k-\
\b-a\^li£ X(J)^2k
-i
and
1 " € b-a+2 , ^
1 £ 1 *-* € w^
Hence
<€
D
for all N>N(t) and this establishes (5.1.2) as required
We remark for the reader's benefit that some authors define discrepancy to be
the function that we have written DN(xn)/N. The function that we have defined is
never less than 1 (take J as an arbitrarily small interval centred on Xj). Indeed,
by a result of Schmidt [243] (see also Halasz [107]), we have
limsUp^^>0<
N->co
log AT
On the other hand, as we shall see in Section 5.3, there exist sequences (xn = ns/2
for example) for which
lim^<C,
N-oo lOg AT
122
Uniform distribution
for some explicit value C. This subject is dealt with in more detail in Chapter 2
of [175].
The following theorem relates normality to base b and the uniform distribution
of the sequence ocbn (Wall [276]).
Theorem 5.2 Let b>\ be an integer. The sequence <xbn is uniformly distributed
(modulo one) if and only if vl is entirely normal to base b.
Proof We shall use the characterization of normality provided by Theorem 1.3
in this proof. Firstly, suppose that a is entirely normal to base b, so that
lim
N-*oo
A(Bk,b,N)
N
= b~k
(5.1.4)
for every block Bk. Let €>0, and let k be an integer such that Ab k<€. Then
there exists N(e) such that for N>N(e) we have, for every block Bk,
A(Bk,b,N) 1
N
<
2b
k-
(5.1.5)
As in the proof of the previous theorem, given any interval J a [0,1), we can find
integers c, d such that
/'
[c-1 d + ll
l~Vr,~bir\i
d—c _,^N d—c + 2
-zr^(/)^—n—=
and, if k(#)>2b fc, as we henceforth suppose,
[cb-k,db~k]<=f.
If g is an integer with 0^g^bk — 1, we write Bk(g) to denote its expansion to
base b. We then have
^AiB.igX^N)^ I U I A(Bk(g),b,N),
g = c n=l g = c-l
{bna}ef
from which it quickly follows, in view of (5.1.5), that
N
X l-NMJ)
7?=1
{bna}e/
<€
and so
DN(bn<x)<€.
Now suppose that (5.1.2) holds with xn = bnai. Let Bk be any block of k digits
(modulo b). Suppose Bk = Bk(d) in the above notation. Then
N
A{Bk,b,N)= £ 1
n=l
Definitions and elementary properties
with f = [db-kAd + \)b~k). Hence
A(Bk,b,N) 1
bk
123
N
^
DN(bn<x)
N
= o(\).
Thus (5.1.4) holds, which establishes that a is entirely normal to base b.
□
Corollary For almost all real <x the sequence ccb" is uniformly distributed for every
integer b^2.
The corollary follows immediately upon combining Theorems 1.4 and 5.2. In
order to investigate other sequences and obtain quantitative bounds for DN(xn),
we shall need to develop some important number-theoretic tools from Fourier
analysis. We close this section with some connections between uniform
distribution and Riemann integration.
Theorem 5.3 Let f(x) be a function of a real variable with period one which is
Riemann-integrable over [0,1). Let xn be a uniformly distributed sequence. Then
1 N
limT? I/(**) =
N-ooiVn=l
n
f(x)dx.
(5.1.6)
Remark Since the characteristic function of an interval is Riemann-integrable, it
follows that an alternative definition of a uniformly distributed sequence is: xn is
uniformly distributed modulo one if and only if (5.1.6) holds for all Riemann-
integrable functions with period one.
Proof Without loss of generality, f(x) is real. By well-known properties of the
Riemann-integral (see [80] Theorem 41, Corollary 3 for example), for every €>0
there are step functions Sj(x) with
Si(x) </(x) < S2(x) for every x
(5.1.7)
and
(Sj(x)-f(x))dx
^€.
(5.1.8)
Now each Sj(x) can be written
Sj(x)= Zc(rJ)X(Jtr,x),
M
r=l
where {J;.r=\,...,M) is a set of disjoint subintervals of [0,1), and x(</r,Jt) is
the characteristic function of Jr. By the definition of uniform distribution,
1
N
n
N-»oo-/Vn=l
Sj(x)dx.
124
Uniform distribution
Hence from (5.1.7) we obtain
1 i N
^(xK liminf-£/(*„)
0 N-oo ™ «=1
1 N
^limsup- £/(*„)<
N-»oo /Vn=l
/•l
52(A:)djc.
Since e was arbitrary, we conclude from (5.1.8) that (5.1.6) holds, as required. □
The following result (known as Koksma's inequality [67]) shows that the
previous theorem can be made qualitative given some bound on the total
variation of/.
Theorem 5.4 Let f(x) be a real-valued function of a real variable with period 1.
Suppose that f has bounded variation V(f) over [0,1). Then, for every sequence xn,
and every integer N^\, we have
N
fix) dx
^V(f)
N
(5.1.9)
Remark The reader will note that our proof shows that (5.1.9) remains true with
DN(xn) replaced by
D%(xn)= sup
0^x<l
I l-Nx
Of course, we have D%(xn) ^DN(xn) ^2D%(xn), as the reader can easily verify.
Proof We may suppose that 0^ x1 ^ x2 ^ ... ^ xN^ 1. Put x0=0 and xN+ x = 1.
Suppose that xm ¥zxm+1 for some m. By considering the intervals [0, ;cm], [0, xm+ J,
we have
DN(xJ^max(\Nxm-m\, \Nxm+1-m\).
If, on the other hand, xn is constant for m—t^n^m + r, we have
DN(xn)^max(\Nxm_t-(m-t-\)l \Nxm+r+1-(m + r)\)
^max(\Nxm-m\, \Nxm+l-m\),
since \A— y\ attains its maximum for j>e[a, /?] at either a or /?. Hence
DN(xn)^ max max(\Nxm-m\,\Nxm+l-m\).
Os£m<N
Now, by integration by parts,
7( *) d* =[*/(*)]$-
xdf{t),
(5.1.10)
Definitions and elementary properties
125
while partial summation yields
N
N
4 Z/w =/(D- Z £(/(*.+i)-/(*.))
7 V n = 1 n = 1 l V
=/(D- I
n — 1 «/ *n
n+1 ft
^d/(x).
Thus
N
n — l
/(*)dx
« = 0 J
n
X~N
dftx)
^ max max
O^n^N x„^x^x„+l
n
X~N
W(x)\
= V(f) max (\Nxm-m\, \Nxm+l-m\}^V(f)
DN(xn)
O^m^N
N
by (5.1.10) as required.
□
We remark that (5.1.9) is essentially best possible, apart from a constant factor.
To see this, let/(jc) be the characteristic function of a subinterval of [0,1). Then,
by a suitable choice, the left-hand side of (5.1.9) is DN(xn)/N, while V(f) = 2. It is
possible to replace V(f) by jV(f) by arguing as follows, and thus make the
inequality sharp. Let
/ = inf/(*), u = supf(x),
and write
*(/,a) = {x6[0,l):/(x)^a}.
If £(f, a) consists of a finite number of disjoint intervals (here isolated points are
considered as intervals), let K(f, a) denote the number of such intervals. Provided
that K(f, a) is defined for all a, write
V*(f) =
*(/,«) da.
It is then a simple exercise (changing the underlying idea of Riemann integration
in the above proof to Lebesgue integration) to show that
V*(f) = iV(f)
and
1 N
m
Ax)6x
^V*(f)
N
126 Uniform distribution
5.2 TRIGONOMETRIC SUMS, THE ERDOS-TURAN THEOREM
AND THE WEYL CRITERION
In order to investigate (5.1.1) we must obtain a means for counting the number of
times {xn} lies in an interval J'. The characteristic function of J has a well-known
Fourier expansion, but this is of limited use since the «th coefficient has size of
order n~l. Vinogradov (see [271], Chapter One) developed an approximation to
the characteristic function of an interval which has proved to be very useful over
the years. We shall make use of a more recently introduced function which
has been described in [18]. As is customary in number theory, we write
e(jc)=exp(27ri;c).
Lemma 5.1 Let xn be a sequence of reals and an a sequence of non-negative reals.
Let J be a subinterval of [0,1) and let N and L be positive integers. Then there are
reals 6U 62 with |0j|<1, such that
«=1 «=1 \ L.Jf\J m=l
N
Z an^rnxn)
(5.2.1)
where
2
c(m) = c(miL,J)=-—- -\-2mm{{nm)~\X{J),\-l{J)).
Proof Let /(«) be the characteristic function of J*, and write
00
X*(x)= £ X(x+n).
n= — oo
For any function f(x)eL1(U) we write}{y) to denote its Fourier transform:
foo
Ky)=
f(x)e(-yx)dx.
— oo
We require an approximation to x(jc), say \j/(x), such that \j/(y) vanishes outside
some interval, say [—L,L]. We then have
L
Z $(n)e(nx)
n=-L
as an approximation to x*(*).
We begin the construction of our approximation by writing
f 1 if x^O
sgn(x) = ^
l-\ if jc<0.
Let / be the closure of J, so # = [a, b] for some a, b, 0 ^a^b ^ 1. Write 0(jc) for
the characteristic function of J'. Then
0(*) = i(sgn(//(jc-a)) + sgn(#(6-*)))
Trigonometric sums, the Erdos-Turan theorem and the Weyl criterion 127
for any positive H. It follows from unpublished work of Beurling (see [249]) that
there is a function F(x) with
F(x)^sgn(;0,
r°°
(F( x) — sgn (x)) dx = 1,
J — 00
and, for any real a, /?,
(F(x-oi) + F(x-P))e(-tx)dx=0
J — oo
when \t\^\. The function concerned is
f(z)=r^]Tf(z_„r2_f(z+„r2+?i
L * J L-0 n=l Z\
The reader can find more details, and the proofs of the results we require, in [18],
[195], or [266]. Now write
u(x)={(F((L + \Kx-a)) + F((L + \)(x-b))).
Then:
u(x)^6(x)^x(x),
u(y) = 0 for[y|^L + l,
and
C°° 1
(u(x)-x(x))dx=—j. (5.2.2)
%) ~~ 00
If we write
oo
w*(jc) = Y, m(jc + /i),
n= — oo
then it follows from a well-known result in Fourier series (Theorem 6.2 in
Chapter 1 of [283]) that, for almost all x,
L
u*(x)= Yj u(n)e(—nx) = v(x), say.
n=-L
We thus have
u*(x)^x(x) f°r aH x,
u*(x) = v(x) almost everywhere,
i;(jc) is continuous.
128
Uniform distribution
It follows that
for all jc. From (5.2.2),
v(x)^x*(x)
w(0) = ;i(JO + (L + l)
-l
Thus
X an^ ^anU(t/) + (L + l)-1)+ t *(»» X ant{-mxn).
n=l n=l
m= —L n= 1
m#0
Now, \u(m)\ = \u(-m)\ and
w(m) =
•oo
- oo
/*oo
w(jc)e( —«jc)d;c
— oo
X(x)e( —mx)dx+
= ^ + 12, say.
'00
(u(x) — x(x))e(—mx) dx
— oo
We have
i/.i-
sin nm(b—a)
Tim
<min
7i\m\
,b—a,l—b+a
Also
r<x>
\I2\<
\u(x)-x(x)\dx =
— oo
L + l
Hence
N
N L
Z «„< 2>„a(jo+(L+iri)+ Zc(m)
n= 1 n= 1 m= 1
We/
N
Z flne(w^n)
(5.2.3)
with c(w) as stated in the lemma hypothesis. By replacing [a,b] with (a,b) and
using
- i(sgn(ff(a - *)) + sgn(i/(x - ft))) ^ *(*),
we obtain, in a similar way,
n=1 n= 1 m=1
{^}e/
Combining (5.2.3) and (5.2.4) gives (5.2.1) as desired.
N
Z an^fnX„)
n=l
(5.2.4)
□
Trigonometric sums, the Erdos-Turan theorem and the Weyl criterion
129
Theorem 5.5 (Erdos-Turan) Let xn be a sequence of reals, and L, N be positive
integers. Then there is an absolute constant C such that
N
1
^XlTT^I^
N
Z &(mXn)
n=l
(5.2.5)
Proof This follows immediately from (5.2.1) and (5.2.2) with an = \, since
2 2
c(m)^-—- +—^w_1(2 + 2/tc).
L + l mn
This gives (5.2.5) with C = (2 + 2/n).
□
It is interesting to note that the first term on the right of (5.2.5) is best possible.
To see this, let xn = (n — \)/N. Then DN(xn) = \, but
N
Ze(wjc„)=0 for l^m^N-\,
n=l
Choosing L=N—\, we then find that (5.2.5) gives \ = DN(xn)^\, there being no
contribution from the second term on the right-hand side.
Theorem 5.6 (The Weyl criterion) A sequence xn is uniformly distributed modulo
one if and only if
1 N
lim- ^eW=0
N-KX1* n=l
(5.2.6)
for every non-zero integer h.
Proof Suppose first that (5.2.6) holds for every non-zero integer h. Then, given
€>0, there is an Af(e) such that
1 N
<€'
for N^N(£), /z = l,2,..., [e-2]. Thus, by (5.2.5),
1
£„(*„) ^Me2+ CAte2 V -^Me
provided that € is sufficiently small. Thus DN(xn)=o(N).
On the other hand, if xn is uniformly distributed, (5.2.6) follows immediately
from Theorem 5.3: indeed, the characterization of uniformly distributed sequences
given there leads to another proof of the Weyl criterion using the fact that
trigonometric polynomials (that is, polynomials in e(nx)) are dense in the set of
continuous functions of period 1. □
130
Uniform distribution
We note that by a careful application of Theorem 5.4 we obtain
N
Z e(/*x„)
n=l
^4\h\DN(xn)
(5.2.7)
for any non-zero integer h, which incidentally provides a lower bound for DN(xn).
To obtain (5.2.7), note that there will be some real number <xe[0,1) such that
I e(/**„)
n=l
N N
= Se(^n + a)= Icos(2rc(k„ + a));
«=i
«=i
and (with V(f) as in Theorem 5.4) V{cos(2%(hx + ai)))=4\h\. The example
xn = (n — \)/N, h=N, shows that (5.2.7) is best possible, apart from the factor 4.
Clearly Theorems 5.5 and 5.6 can be generalized to deal with weighted
discrepancy or uniform distribution. For a non-negative sequence an we could
write
DN(xn,an)= sup
./c[0,l)
N N
YJan-X{J)Yjan
One particular example is afforded by the choice an=n~1, which produces the
so-called logarithmic discrepancy of a sequence. In this situation it is no longer
efficient to use the same approximating function for each n (which is natural, since
the points no longer carry equal weights). An Erdos-Turan type of theorem in
this context has been given as follows [19].
Theorem 5.7 Let <5e(0,1). Then for every sequence xn we have
DN(xn,n-1)<C(d) + 24 X %ax
N j
I -e(/**„)
n = An
where C(<5) depends only on S.
A sequence will be logarithmically uniformly distributed modulo one if and
only if DN(xn,n~ 1)=o(\ogN).
We close this section with some simple applications of the results, before we
move on to discuss the metric theory. The subject of uniform distribution began
with the study of {not} where a is an irrational number. It is elementary to prove
that this sequence is uniformly distributed modulo one (see the next section).
Another proof follows from the Weyl criterion. We have, by the formula for the
sum of a geometric progression, that
N
Z e(hncc)
n=l
<
1
1
|e(/za) —1| \siri7t<xh\ ||a/z||
If a is irrational, then || ah || is non-zero for h #0. Hence an is uniformly distributed,
by the Weyl criterion. It is possible to bound DN(<xri), by Theorem 5.5, if something
is known about rational approximations to a. Roughly speaking, if a does not
The metrical theory of uniform distribution 131
have very good rational approximations, then DN(cm) will increase slowly. This
follows from (5.2.5), since we obtain
N L
~ -'l/fiair1.
-k+1 m=l
In practice, the result obtained from the Erdos-Turan theorem will be worse than
that obtained by the elementary argument of the next section for this sequence by
a factor logiV.
As an example of a sequence which is not uniformly distributed modulo one,
although its fractional parts are dense in [0,1), we mention logw. Here,
N N \r2ni +1 _ i
Ie(log/!)= 5>2ld = ——.— +0(1).
„=i „=i 2m
Thus (5.2.6) fails. However, the sequence is logarithmically uniformly distributed,
and therefore dense in [0,1), as can be seen from Theorem 5.7 with 5=\ say.
Indeed, we have
Djflogn, n~')<K
for some constant K.
The fact that log« is not uniformly distributed modulo one is an example of
the following result [264]: Suppose that an is non-decreasing and an = 0(\ogn). Then
ancc is not uniformly distributed modulo one for any a. To see this, first note that if
an^K\ogn and is non-decreasing, then, given rj with 0<rj<j, there are infinitely
many values N and real numbers /? with
P^a„^P + rj for N(l-c)<n^N, (5.2.8)
where c = y\12K. If one takes rj = (2 +1 a|" *), it follows that there are infinitely many
intervals (mod 1) and infinitely many N such that at least cN consecutive values
up to N all lie in an interval. Hence DN(anaL)^cNI2, and so DN^o{N). To prove
(5.2.8), argue by contradiction: the vital fact is that — ATlog(l — c)<rj. We have
dwelt on this example because we shall consider slowly growing sequences from
a metric perspective in the next section.
Many more sequences of the form xn=f(n), where / is some function whose
derivatives have suitable bounds, can be investigated by Theorem 5.5; see Chapter 2
of [175].
5.3 THE METRICAL THEORY OF UNIFORM DISTRIBUTION
In this section we shall be concerned with the uniform distribution of {ana} for
almost all a, where an is some given sequence. We mention at the outset that if an
is a sequence of integers, then the problem is periodic in a. In this case we need
only consider ae[0,1). If an is only known to be a real sequence, we might consider
ae[A,A + l], for some A.
132
Uniform distribution
Alternatively, in some proofs it is useful to work with the measure pi defined
on the c-algebra of Lebesgue measurable subsets of R by
ti*) =
7r1/2exp(-7rV)d2.
(5.3.1)
s
The measure \i has the following three useful properties:
H(U) = \;
iu(^) = 0<=>A(^)=0;
e( — olx) dfi=exp(—jc2) .
xeU
The last of these properties is of course the Gauss integral formula. By the second
of these properties, 'almost all' with respect to \i is equivalent to 'almost all' with
respect to Lebesgue measure.
Another complication arises when we deal with real sequences, namely that
we need some control on the size of \an—am\ for m^n (we no longer have the
useful property that distinct integers are spaced at least 1 apart!). We begin with
a very simple result.
Theorem 5.8 Let an be a sequence of distinct integers. Then the sequence an<x is
uniformly distributed for almost all a.
Proof By ParsevaPs equality, we have, for every h^O,
n
Z Q(han*)
n = u
doc = v —w + 1.
Hence, by Lemma 1.4 or Lemma 1.5, for any given non-zero heZ, for almost all a,
lim-
N-ooly
N
Z e(H«)
n=l
=0.
(5.3.2)
To obtain this result we need to treat the real and imaginary parts of the
exponential sum separately and take, for the real part, fn(x) = \ + cos(2nhanx),
f„ = (pn = \. The imaginary part is treated similarly, with sin replacing cos. Since a
countable union of sets of measure zero itself has measure zero, (5.3.2) holds for
some set with full measure for every h^O. The result then follows by the Weyl
criterion. □
Clearly, in the above we could have relaxed our condition that the an should
be distinct, that is to say, some repeats could be allowed. Before we prove a
stronger form of the result which not only allows for repeats amongst the an but
also allows non-integral values for the an, we establish a more general result which
has many corollaries.
The metrical theory of uniform distribution
133
Lemma 5.2 Let X be a measure space with measure pi such that fi(X)< oo. Let
f(n, a) be a sequence of \i-measurable functions such that
|/(«,a)|<K for some K>0,
and
oo j
^ ~n*
I Am, a)
m=l
d/i
converges. Then, for almost all a, we have
1 N
— £/(«,a)->0as 7V->oo.
■'Vn=l
Proo/ First we note that we can find an increasing sequence of reals o(ri) such
that
<t(1) = 1, <r(2ri) < 2<r(w), ain) ^ n, a(n) -* oo as « -»• oo,
and
2^ M3
n=l
JT
X /(w> a>
m=l
djU
converges. Clearly we may extend o(ri) to be a continuous increasing function of
a positive real variable jc with <r(x)^x. Write
R 12
m=l
7(«) = (72(«)
We define 0(r) and K(r) inductively by:
0(0) = 1, 0(r+l) = 0(r)
1 +
—1
K(r) = 0(r+l)-0(r).
It then follows that K(r)^2, 0(r+l)<30(r). For r=0,1,2,..., pick N=N(r) such
that 0(2rKJv"<0(2r+1), and
/(AQ /(m)
AT2 ^ m2
for0(2r)^w<0(2r+1).
Then
- /(Mr)) - 1 2
2-, Ar/«\2 ^ 2- \T(V\2
I /(/I)
" 1 (7(0(r))
= 2. 777^2 -B73T I /(h)
oo
■ -! H3
134
Uniform distribution
If we write
we thus have
00
I
r = 0
\S(N(r),oL)\2dfi<oo.
Thus, by Fatou's lemma, we have
£|S(tf(r),a)|2dA«<oo,
and so
X r = 0
oo
X|5(iV(r),a)|2<oo
r = 0
for almost all a (say cleS). Now let ae<f; then there exists #(a) with
|5(^(r),a)|<jy(a).
Hence, given M, we can find r such that JV(r)^M<JV(r+l), and
i M
1*1 „=1
\ N 1 M
-—-S{N{r\0L) + - X /(/i,a)
^ ► () asM-»-oo.
<7(A0
This completes the proof.
□
Combining the above lemma with the Weyl criterion (and the fact that a
countable union of sets with measure zero has measure zero), we obtain the
following theorem of Davenport, Erdos, and LeVeque [69].
Theorem 5.9 Suppose that a, b are real numbers with a<b, and fn(x) is a sequence
of real functions such that
00 J
N=liy
rb
N
I e(/*/„(a))
n=l
da
converges for every positive integer h. Then the sequence /„(a) is uniformly
distributed modulo one for almost all ae [a, b].
The following result leads to many corollaries in conjunction with the above
theorem.
The metrical theory of uniform distribution
135
Lemma 5.3 Suppose that f(x) is a real-valued function with a monotonic derivative
for xe [a, b], which satisfies \ f'(x)| ^ X > 0. Then
Cb
e(/(a))da
^ nX
Proof Without loss of generality, /' is positive and non-decreasing. There exists
0with
Cb
e(/(a))da
rb j {
—d(cos(27^/(a) + 0))) = -
rb
cos(2rc/(a) + 9) da
d(cos(27r/(a) + 0))
a/'(a) 'w f\a)
(for some ce(a,b) by the second mean-value theorem for integrals)
1 sin(2;t(/(c) + fl)-sin(27r(/(a) + fl)
~f(a) Ik '
from which the result immediately follows.
Combining Theorem 5.9 with the above gives the following.
□
Theorem 5.10 Let fn(x) be a sequence of functions such that f'n(<x) — f'm(ct) is
monotonic for each pair w, n, for ae [a, b]. Write
jS(m,H) = min|/>)-/;fa)|.
Then the sequence /„(a) is uniformly distributed modulo one for almost all <xe[a,b]
if the following series converges:
oo
£AT3 X mina^m,*)-1).
N=l myn^N
Proof We have
rb
N
I e(/*/n(a))
n=l
rb
da= X e(/z(/n(a)-/m(a)))da.
The result then follows from Lemma 5.3 and Theorem 5.9, using the trivial bound
for the integral when fl{m, n) < 1. □
Remark An alternative hypothesis is clearly the covergence of
f logiv;
L N3 L *»
N=l iy m,n^N
0(m,n)<l
136 Uniform distribution
since a little manipulation yields
X min(l,i5(m,«)-1)« jj J 1.
m,n^N h=l "■ m.nsgN
P(m,n)<l
This leads to the following immediate corollary.
Corollary 1 Let fn(cc) be a sequence of functions such that f'n(<x) —f'm(p) is monotonic
for each pair m, n, for ae[a, b]. Then if
\fJL*)-fJL*)\>8form¥>n,
for some S>0, the sequence fn(cc) is uniformly distributed modulo one for almost
all a.
Corollary 2 The sequences ccn and <xbn (where b is some given real exceeding 1) are
uniformly distributed modulo one for almost all <x > 1.
Proof In the above notation, for the sequence a" and for n>m we have
P(m,n) = <xm~1\noLn~m-m\^n-m.
For the sequence cdf we have p(m, n) = | bm—bn | ^ b — 1. □
Remark We note that the Corollary to Theorem 5.2 dealt only with <xbn where
beZ, b^ 2. Clearly we can also deal with afl" where an does not have to increase
very quickly, indeed an+l— an> (log ri)2+€n~1 suffices. Also we can deal with aan
under the same hypothesis.
We are now in a position to establish an improved version of Theorem 5.8
which is a stronger result than can be obtained from Theorem 5.10 (we save a log
factor by using the measure fi).
Theorem 5.8* Let \j/(x) be a non-increasing function taking positive values for
x ^ 0 and such that
\J/(x)dx (5.3.3)
o
converges. Then the result of Theorem 5.8 holds for real sequences an provided that
£ l««^(log2«).
K-*J<1
Proof We work with the measure \i introduced in (5.3.1). We have
xeU
N
Z e(^na)
«=1
2
d/i= £ exp(-h2(an-am)2)
^N+2^ £exp(-/z2(*:-l)2) £ 1
n=lk=l \am-aj*k
m<n
The metrical theory of uniform distribution 137
N oo
«n+y, 2>exp(-/*2(A:-i)2)wKiog2w)
n=lk=l
N
«N+ J]«^(log2«).
The proof then follows from Theorem 5.9, since
00 1 N °° 1 °° il/(\os2ri)
n=l-/V „=1 n=1 N^n1* n=l n
r<x>\J/(\ogx) rco
«
d;c =
i x
\j/{x) dx. D
0
Remark We note that if an>an_ x + c(logw)1 +dn~l, then a„a is uniformly
distributed for almost all a. On the other hand, in the previous section we proved that
it an <an_ j + cn~1, then anoc is uniformly distributed for no a. We shall demonstrate
that Theorem 5.8* is best possible, since under the same hypotheses, but with
(5.3.3) divergent, there exists a sequence an such that anoi is uniformly distributed
only on a set with measure zero (see Theorem 5.12, Corollary 2).
In order to establish quantitative versions of the above results, we require the
following variant of Lemma 1.5. The proof is almost identical, so we shall omit
the details.
Lemma 5.4 Let X be a measure space with measure pi such that 0 < n(X) < oo. Let
F(n, w, a) (n = 1,2,...; w = 0,1,2,...) be a double sequence of \i-measurable
functions, and let cpn be a sequence of real numbers such that
\F(n,n-\,<x)\^(pn (« = 1,2,...).
Write
<W0=X>„
7?=1
and suppose that 3>(oo) diverges. Also suppose that for arbitrary integers u,v
(0^u<v) we have
r v
{FiUiViOOfdiKK^Vn (5.3.4)
( n = u
for an absolute constant K. Then, for almost all ol, we have
F(N, 0, a) = 0(<D1/2 (N) (log (O(A0 + 2))3/2 +€ + max q>n).
Theorem 5.11 Let an be a sequence of distinct integers and € > 0. Then, for almost
all a, we have
DN(anaL)« Nm(\og 2N)5'2 +e.
(5.3.5)
138
Uniform distribution
Proof We let
v 1
F(u,v,a)= I t
I e(^na)
«=«
Then
" 1
fl w
|F(M,t;,a)|2da^ It It
0 h=in JO k=lK
1
I e(/w„a)
i; 1 \ 2
da.
-(—+«(tij)
^4l(log2«):
n = «
Hence, by Lemma 5.4 and the Erdos-Turan theorem, we obtain (5.3.5) as
required.
The above result was obtained simultaneously by Cassels [55] and Erdos and
Koksma [94] in a more general form which we shall consider later. The term N112
in the theorem is best possible, as can be seen by various examples. By a result
of Behnke [27], for any a we have DN(<xn2)»N112 for infinitely many N. It was
shown by Fiedler, Jurkat, and Korner [97] that, for almost all a,
SN(<x): =
I e(a«2)
n=l
= 0(Nll2g(N))
or
SN(0L)=Q(Nll2g(N))
according as the series
00
1
^ng\n)
converges or diverges. Hence DN(<xn2) = Cl(N1/2\ogll4N) for almost all a, for
example, by Theorem 5.4. This disproved a conjecture of Erdos [91] that
DN(ccan)=0(N1/2\og\ogcN) for some c, for any increasing sequence of integers an,
although this was not noticed for nearly twenty years [262].
We note that it follows from our results in Section 1.3 that, for almost all a,
Z>N(2na)>cAT1/2(loglogA01/2
for any c<2"1/2 and infinitely many N. In fact, one can obtain an exact iaw of
the iterated logarithm' for bn<x; see [215]. By a more recent result of Berkes and
Philipp [29], the exponent of the logarithm in (5.3.5) cannot be reduced below \.
This is an immediate corollary of the following theorem, which constructs bad
sequences. We here show that the good distribution of lacunary sequences in short
The metrical theory of uniform distribution 139
intervals (guaranteed by Theorem 3.1) forces certain other sequences to have a
large discrepancy because there are too many fractional parts in large intervals.
Theorem 5.12 Let \l/(x) be a positive function of a real variable x such that
\j/(3 log x)x~* is non-increasing for large x, \J/(x) ^ exp (—xl3), and (5.3.3) diverges.
Then there exist increasing sequences of integers an such that for almost all cc there
are infinitely many N with
||fl,a||<J for N^n^N+Nll2\J/-ll2(\ogN). (5.3.6)
Proof Let bn be a lacunary sequence with bn > (n2 +1) bn _ 1. By Theorem 3.1 there
are infinitely many solutions to
||^a||<(5«)-V(31og«) (5.3.7)
for almost all a. Write r=r(m) = [m\J/~1(3\ogm)] + \. Let J>m be the finite sequence
omt ^om, jOm,..., rom.
Then if (5.3.7) holds for n=m, we have ||af||<£ for teJm. Pick an as the
consecutive members of Jx, J2,... By our hypothesis on \J/ we have r ^ m2 +1,
and so an is an increasing sequence. Now suppose that (5.3.7) holds for n=M.
The number of members in the sequence an included for m ^ M is
X wiA"1(31ogw)^MV"1(31ogM).
Let the first member of JM, that is, bM, be the Nth. member of the sequence an.
Thus N<M2\l/~x(3logM) (in particular, N^M3) and
Ka|| <i for N^n^N+r(M).
The proof is completed by noting that
r(M)^(M2\J/-\3\ogM))ll2\J/-ll2(3\ogM)^Nll2\J/-ll2(\ogN). Q
Remark Clearly the condition \J/(x)^exp(—xl3) could be relaxed to \J/(x)^
exp( — (1 — €)x) for some €>0. Several variants on the construction are possible.
For example, one may take bn = 2n, and let Jm be the sequence bm, 3bm, 5bm,...,
which gives a sequence of distinct, but not increasing, integers an. The sequence
chosen by Berkes and Philipp was bm = 2m2~m+l, which comes within our family
of sequences, but they did not work via Diophantine approximation as we have
done.
Corollary 1 Given the hypotheses of Theorem 5.12, there are sequences an such that
for almost all a there are infinitely many N with DN(an<x) ^ Nll2il/~1/2(\ogN).
140 Uniform distribution
Corollary 2 Let <p(x) be a positive-valued, non-increasing function of a real
variable x such that: x~1^ q>(x) ^x~2, q>(2x) ^ 6<p(x) for some d >0, and
(p(x)dx
o
diverges. Then there exists a non-decreasing sequence of positive integers an such that
£ \^nq>(\ogn)
and for almost all a the sequence ana is not uniformly distributed.
Remark The conditions on q> can be relaxed somewhat. Those we have stated
include the most interesting cases. Corollary 2 was first proved by R.C. Baker [9]
nearly twenty years before the result of Berkes and Philipp, who, although noting
that there was 'some similarity' between their construction and Baker's, did not
notice that the same 'bad sequence' could be used for both results.
Proof of Corollary 2 Write 6(x) = xl(p(x). Thus 6 is strictly increasing and hence
its inverse, say £, exists. By the hypotheses on (p we have x2 ^ 6(x) ^ jc3 and
xll2^£(x)^x113. We can then define a function \J/(y) by \J/Q\ogy)=y<p2(l;(y)).
We note that \J/(3 log y)/y is non-increasing, \j/(y) ^exp(— yl3), and
Hy)dy=3
3
roo
*(31ogr;
r<x>
dt = 3
t
(for some value c using the change of variables t = 8(x))
q>2(x)d6(x)
>3
" 2/ . d*
<p\x)
q>(x)'
which diverges, by our hypothesis. Thus \j/ satisfies the conditions of Theorem 5.12.
Also, il/(3x)»\J/(x\ where here, and elsewhere in the proof, implied constants may
depend on S. Hence there is a sequence bn such that (5.3.6) holds for almost all a
and infinitely many n with an=bn. We now insert repeats into the sequence as
follows. Pick al=b1, a2=b2, and in general suppose that au...,aj_1 have been
picked from b1,...,br_1. Then let
aj=aj+1 = ...=aj+h = br, where h=j(p(\ogj).
Let /= 0(logy). Thus J«r«J, by our hypotheses. Hence if (5.3.6) holds for br and
r1/2(/f~1/2(logr) consecutive values, then ||a„a|| <\ for n—j and
»j(p(\og(j))J112^ ~ 1/2(log (/)) »M\og(j))J112^ ~ 1/20 log (/))=/
consecutive values. Hence anoi cannot be uniformly distributed. □
The metrical theory of uniform distribution
141
We now consider Baker's improvement of Theorem 5.11 [17], which shows that
the exponent 5/2 could be replaced by 3/2 by using the following result of Hunt
[140].
Lemma 5.5A Let cn be a sequence of complex numbers such that
oo
Ikl
converges. Then
n=l
max
0 N^l
N
Z Cn^0C)
oo
da<*J>J
n=l
where K is an absolute constant.
Montgomery and Vaaler [196] have shown that Lemma 5.5A is equivalent to
the following result.
Lemma 5.5B Let cn be as above, and suppose that an is a sequence of reals with
an+l—an^5>0 for \^n<N. Then, for any T>0, and any T0, we have
rr0+T
max
m
Z VfrwO
n=l
N
da^tfor+^XW
(5.3.8)
n=l
The optimal constants K in Lemmas 5.5 A and B are identical.
Now write
N
SN(oc)= Ze(a„a).
n=l
It then follows from (5.3.8) that, for any /z^O and every M^ 1,
CA+l
max |SN(a/z)|2 da ^ KMb~1.
(5.3.9)
A N^M
The presence of the maximum inside the integral enables us to produce a
stronger conclusion than can be obtained from Lemma 5.4, as we now show. Let
lK«)=H1/2(l0g«)3/2+€.
Thus
,/,(4*+i)<4,/,(4*)
for all sufficiently large k. Hence, if
max/)Ar(ana)<i(A(4k),
(5.3.10)
we have
DN(ana) <if/(N) for 4* "l ^ N ^ 4
142
Uniform distribution
We therefore need only show that (5.3.10) holds for almost all a for every
sufficiently large k in order to prove the following.
Theorem 5.13 Let an be an increasing sequence of reals with an+l—an^3>0, and
let € > 0. Then, for almost all a, we have
DN(anx) «N1/2(\og2N)3>2+<. (5.3.11)
Proof We have
max/)N(ana)^3(2k+ £ -m2ix\SN(oih)\
by Theorem 5.5. Thus, using (a + b)2 ^ 2(a2 + b2), we obtain
rA + i
max DN(anoc)2 da ^ 18(4* + k24k)K6 "1
J A N<4*
by (5.3.6) and the Cauchy-Schwarz inequality. Hence the measure of the set Sk
for which (5.3.10) fails is «K~1~2'. Thus, by Lemma 1.2, almost all a belong to
only finitely many Sk. Hence (5.3.11) holds for almost all <xe[A,A + \]. Since A is
arbitrary, this establishes the result.
Theorem 5.13 was proved for integer sequences by R.C. Baker [17]. The result
as stated above has not appeared in the literature previously. It is not entirely clear
which, if either, of Theorems 5.13 and Corollary 1 to Theorem 5.12 represents the
truth. The appeal to Erdos-Turan in Theorem 5.13 might be expected to introduce
an extra log factor, which suggests that Theorem 5.12, Corollary 1, is best possible.
Unfortunately, the appeal to Hunt's work seems to rule out replacement of the
sequence a„<x by /„(a) where /„(a) is a sequence of differentiable functions with
l//(a)-/m'(a)l^<5>0 form**.
(5.3.12)
This was the more general situation considered by Erdos and Koksma and by
Cassels, which we prove in the following.
Theorem 5.14 Let fn(x) (n = 1,2,...) be a sequence of continuously differentiable
functions such that (5.3.12) holds for all xe [X, Y] for some Y > X. Then, for almost
all ae[A\y], we have Z)N(/n(a))«iV1/2(log2A05/2+€.
Proof By Lemma 5.3 we have for m#«, h ^ 1 that
e(/z(/n(a)-/m(a))da
It follows that
m
Z e(/z/„(a))
n = u
< max h l\fn(x)-fm{x)\
xe[X,Y)
= j9(m,«), say.
daO—M + l + 7 Y P(m,n).
h
-1
The metrical theory of uniform distribution
143
Now, for any given x we can order ffiat) as follows:
AC*) </;(*)-<5 ^fn3(x)-2S ^ ^/;(x)-(r-l)<5,
(5.3.13)
where u^nj^v, r = u — v+\. At first sight it might appear that the choice of ni
depends on the x chosen, but since
\f*jLx)-fm,Jx)\>S for all xe[X,Y]
and the derivatives are continuous, the ordering (5.3.13) is preserved for all
xe[X,Y]. Hence
£ j9(m,«)«<5-1rlog(2r).
u^m<n^v
Let
v 1
F(v,u,cc)= X t
Z e(/z/n(a))
n = u
Then, by the Cauchy-Schwarz inequality,
rY v i py v i
|,F(z;,M,a)|2da< S- £ -
X h=inJ Xk=lK
I e(£/„(a))
n = ii
da
«(log2w)£
h=l
V—U+l 1 ^
«(l7 — M+l)l0g22«.
The result then follows by Lemma 5.4 and Theorem 5.5.
□
The conditions of Theorem 5.14 are met by many types of sequences, for
example exp(wa)(a>0), a" (a>l), na (a>l), or a0" (0>l). Paradoxically, we may
know nothing about the set of 'almost all' numbers for some of these results, but
know some types of numbers which belong to the exceptional sets (for example,
if a is a PV number [220], then a" is not uniformly distributed modulo one, since
||an||-»0 as w-»oo). The question as to whether e" is uniformly distributed is a
notorious unsolved problem. In Theorems 5.10 and 5.14 one can replace the
hypothesis involving first derivatives with one involving second (or higher)
derivatives, using the result:
rB
e(/(a))da«A_1/fc,
where X is the minimum absolute magnitude of the A:th derivative. Clearly the
conclusion of Theorem 5.14 would have to be suitably weakened. For example,
with the second derivative replacing the first we would get
DN(fn(z))«N3!*(\og2N)5'2+<.
144 Uniform distribution
With more work, one can deal with certain sequences of functions, for example
a^cosa^ where an is an increasing sequence of integers, for which no suitable
hypothesis on the derivatives holds (see [76], [203]).
We now turn out attention to the sequence no.. The following elementary result
is a more useful tool than the Erdos-Turan theorem in this context.
Lemma 5.6 Let cube a real number and qj(j = 0,\i...) an increasing sequence of
integers such that for some sequence a} we have
a-
i
a—-
aj
<-x with(a,,^) = l. (5.3.14)
q)
Then
n / n^? v min(JV,g+1) ,OK,
DN(not.)^3\ l—, (5.3.15)
j-o ai
where r is given by qr<N^qr+l.
Proof Clearly we can find non-negative integers t3 such that
j=o
and 0 ^ tj; ^ min (N, qj+ J qjl. Now let f = [a, /?]. Then the number of solutions to
{nix}ef with A <n ^ A + q}
is no more than the number of solutions to
qfL — \^{n—A)ai+Aaqi—mq^iqfi + \, A<n^A + q}, meZ.
It follows that (n — A)aj can take on at most #,(/?—a)+ 3 values (mod qj). Since
(apqj) = \, this gives an upper bound
qj(ft-a) + 3.
for the number of solutions. Similarly, if £ = (a, /?), we get a lower bound
qj(p-a)-3.
Thus, for any interval J of length X, we have
£ \-NX=ttjqjX + 3ettj-NX = 3dttp
n=l j=0 j=0 j=0
{na)eJ
where \0\ ^ 1. The bound (5.3.15) follows immediately. □
Now if we let qs be the denominators of the successive convergents to the
continued fraction expansion of a, we know from Theorem 4.5 that
£^«rG(logr)
i=o aj
The metrical theory of uniform distribution 145
for almost all a, where G(x) is any positive non-decreasing function such that
IW1
n=l
converges. For any a we have q^^q]1 ^2, so that
max{y: qi ^ N}«log N.
We therefore obtain the following result of Khintchine [153].
Theorem 5.15 For almost all real a we have
DN(n<x)«log IN G(loglog 9N) (5.3.16)
for any positive non-decreasing function G(x) such that
oo
converges.
The above result is best possible in view of the following.
Theorem 5.16 For almost all real ol we have
llmSUP 1 xrrr/i 1 ZR > 0> (5.3.17)
n-oo logAT/f(loglogAO
where H(x) is any positive non-decreasing function such that H(x) ^ e* and
diverges.
Proof We have proved in Chapter 2 that almost all a have infinitely many
approximations of the form
a=-+ 2 _rr/1._1._ _x, (<*,?) = !,
a 0_
# ?2log#i/(loglogtf)
where |0| ^ 1. Put A/^=5^1og^if(loglog^). Then consider the interval
M^q,
Clearly {nu)$# for n ^ N. Thus
JV 1
£*(«<*) > — = - log q #(loglog q)»log Af #(loglog N),
which completes the proof. D
146
Uniform distribution
We note that the exceptional set in the above theorem includes all quadratic
irrationals, since for these the partial quotients are bounded and hence DN{ncn)«
\og2N by Lemma 5.6 (see [210]).
We now return to the connection between sums and integrals that we began
to study in Section 5.1. By combining Theorem 5.4 with Theorem 5.13, we obtain
the following result.
Theorem 5.17 Let fix) be a real-valued function of a real variable with period one.
Suppose that f has bounded variation V(f) for xe [0,1). Let an be a sequence of
distinct integers (or reals satisfying an+1^an + d for some d > 0). Then, for almost
all <x, we have
n — i
n
fa)dt
«V(f)N-1/2(\ogN)3/2+<
(5.3.18)
Similarly, combining Theorems 5.15 and 5.4 produces the following theorem.
Theorem 5.18 Let fix) be as given in Theorem 5.17. Then, for almost all cc, we have
n — i
n
fit) At
« ^P log IN G(loglog 9N),
(5.3.19)
where G(x) is any positive non-decreasing function such that
converges.
Both of the above theorems are essentially best possible, as can be seen by
taking/as the characteristic function of a subinterval of [0,1) extended to have
period one. So far we have dealt only with functions of bounded variation.
Marstrand [190] has shown that the result
1
N
n
N-ooiVn=1
f{t) At
is false for almost all a, for certain sequences an (including an=n) and for /the
characteristic function of certain measurable sets. We shall prove one case of his
theorem as Theorem 5.20 below. We therefore need some condition on the
'niceness' of/ in order to proceed. If we wanted a more general result, it would
therefore be natural to involve the Fourier coefficients of fix). Koksma [170]
showed that, if an is a sequence of distinct integers, then
1 N
n
fix)dx
(5.3.20)
The metrical theory of uniform distribution
147
when
I|cJ2log|A:|
fc#0
converges. Here ck is the A:th Fourier coefficient of f(x). Erdos [88] had earlier
proved a stronger result for lacunary sequences, namely that the convergence of
5>fc|2(loglog(|A:| + l))2
sufficed. We can use the results of Chapter 3 to prove the following.
Theorem 5.19 Let /eL2([0,1)) be periodic with period 1, and ar a sequence of
distinct integers. Suppose that f{y) has Fourier coefficients ck and that, for some
<5e(0,l] and all w#0,
Lkj2«W~*-
fc#0
(5.3.21)
Then, given € > 0, for almost all a,
1 N
ly n=l
PI
AOdt
«C(0L,an,€,f)N-8'2+'.
(5.3.22)
Proof We have, by Parseval's formula,
n
£ f(antx)-(v-u)
n
0 u<n^v
AOdt
da= X ckCm-
ank = arm
u<ny r^v
Now if ank = arm, then k = hs(n,r), m = ht{n,r), where /z is a non-zero integer and
s(n,r) = ar(a„ary\ t(n,r) = an(an,ar)~1.
Thus
2-f C*Cm — 2-f 2-f C/isC/if
anfc = arm
u<ny r^v
u<ny r^v h^O
«
c^yy12
^ x \a a \
by the Cauchy-Schwarz inequality, (5.3.21), and the definition of s and t. By
Holder's inequality we have
'(a* ar)
2X5/2
<
^ \ \a a \
I 7—n75 )(»-")
2-25
«<n,r^ V l^rfln
1/2
^(i?-k)2~*+€/2
148 Uniform distribution
by Theorem 3.2. Since
u<n^v
we may appeal to Lemma 1.5 to complete the proof. □
We finish this section by considering the following conjecture of Khintchine:
If $ is any measurable subset of [0,1), and % its characteristic function (extended
to have period one), then, for almost all a,
1 N
lim-£x(na) = A(*). (5.3.23)
JV-oo VVn=1
The formula (5.3.23) is true for all measurable sets £ if n is replaced by qn, where
q is any integer exceeding 1, as was shown by Raikov [227]. The falsity of
Khintchine's conjecture was established by Marstrand [190], as we now show.
Theorem 5.20 Given €>0, there exists a measurable subset $ <=[0,1) such that
l(S) ^ e, yet, for every real a,
1 N
limsup- £x(«a) = l.
AT-oo TV n=1
Remark The construction can be varied to allow certain other sequences to
replace n (see [190]).
We need some notation and one lemma before we start the proof. Write pin)
for the largest prime factor of n and put
jj/(w) = {»eZ: p(n)^m}, s=s(m) = n(m),
Ks= -} 11 flog/*)- \ U(s, x, sf) =sKs(\ogx)s- \\ogA).
S' P^m
Lemma 5.7 For fixed real m ^ 2, A > 1, s ^ 2, we have
\<srf(m)n[x,xA]\^U(s,x,A) asx->-oo. (5.3.24)
Proof Let Pj denote they'th prime. Clearly we need only to take m=ps. We then
establish (5.3.24) by induction on s. When 5=2, we can use the uniform
distribution of {noc} with a = (log3)/(log2) to show that
|{(m,W):x<2"3"<^}|a(log'3°^og2)logx.
For s > 2, we have
The metrical theory of uniform distribution 149
If we now assume, as an induction hypothesis, that (5.3.24) holds when s is
replaced by 5—1, then the inner sum has the asymptotic value
(5-l)tf,_1a<«i4)aogx-fllogA)'-2.
Comparing a sum with an integral we find that
£ (\ogx-a\ogpsy-2= ±-±-{ + 0((log*;r 2),
a < (log Ax/log ps) \S l)l0S>Ps
so that the inductive step follows from the definition of Ks. This completes the
proof. □
Proof of Theorem 5.20 We first prove the result for almost all ae[0,1): the
extension to all a is very simple. We shall construct two increasing sequences of
integers an,bn with associated functions \l/(ri), ty'in) such that there are infinitely
many solutions to
\ctan—m\<if/(n), (m,«) = l (5.3.25)
for almost all a, say ae J*\ On the other hand, the set £ of a for which there
are any solutions to
\abn-m\<\l/'(n), (m,w) = l (5.3.26)
satisfies A(£)^€. Our construction will show that if ae#", then there are infinitely
many pairs (B, q) with B ^ 1, #-»• oo, such that no. e £ for B ^ n ^ Bq. This will com
plete the proof. To go from #" to the whole of IR, simply take the union of £ with
[0,l)nQ \Jn-\® + m),
n=1m=1
where ^ = [0,1)\#\ Thus, if ae^, then {wa}e<f for all n. Since ^ has measure
zero, and a countable union of sets with measure zero still has measure zero, we
have not increased the measure of £.
We begin our construction by producing two sequences of finite sets of integers
Jp /j. The an will be the members of Ji in ascending order, while we obtain the
bn from the fy Let J ^ =■# \ = {2}. Now suppose we have constructed the sets for
j < k. Let the largest member oiJk_x be Y, and write Ik_ x = | Jk_ l |. Let m=kk+2,
w=m!, and
3tf{x) = vvj^(m) n [x, kx], &(x) = w2stf(m) n [A:*, mx].
By (5.3.24), as jc-»oo,
|^(x)|^t/(5,x,A:), | ^(x)|£ (A: +1)17(5, x,Jfc).
Thus, we can find an x>Y(k — \)k~1Ik^1 such that
k\Jf(x)\<\9(x)\.
150 Uniform distribution
We now take Sk = y(x), fk = 34?(x). Also, if aneJk, we let \J/(ri) = r]Ik1, and if
bnefk, then \J//(ri) = rjIk1, where r\ will be chosen later in terms of €. We note that
if an e Jk and br e fk, then
cp(an) _<p(br)= nA_rue_? - e_v
a„ K pVmV />/ logm (k + 2)\ogk
Here (p(w) has reverted to its usual meaning as Euler's totient function, y is Euler's
constant, and we have used Theorem 429 of [111] (Mertens' theorem). It follows
that
oo
diverges, while
„=1 an
Innm
n= 1 un
converges. By a suitable choice of rj we can therefore make 2(<f)^e. If we knew
that the Duffin-Schaeffer conjecture were true, we would be able to conclude that
there are infinitely many solutions to (5.3.25) for almost all a, but instead we need
to do a little more work. Write
•.-non u (tmti±m\
Then, ifa„,areSk,
A(@n n @r)« X{0&n) l(@r)(k + 2) log k,
by (2.4.1). On the other hand, if arejj withy <&, then
an^(r) ajjr) _ _^_ ,k+2
(an,ar) ar I}ar
by our construction. Since all the primes in P(r, n) in (2.4.2) must divide an, and so
not exceed kk+2, it follows from Lemma 2.8 that X(0$nn@^«X(@n)X(0$r). Hence, if
then
X l(^n^r)= X Z I ^-^^r)
l^r<n^N 7=1 m=1 ane./,.
«( Z ^(^-)Y+ Z /J(//7 + 2)log7)-2(7 + 2)logy
« f Z ^(^n)Y,
Uniform distribution in higher dimensions 151
since
k g-y
„?/Ws2*£o>2).ogy
Hence, by Lemma 2.3, there are infinitely many solutions to (5.3.25) for almost
all a.
To complete the proof we need to establish that there are infinitely many pairs
(B,q) with {ha}GS for ae^ B^h^Bq. Suppose
fj-Wn) .7+ *(*)'
OLE
an an
where aneJk. Then
V ajh ' ajh
If we can show that anh~le#k> then {hoi}e£ by our construction. Let
B=a„(xk)~1 (here x has the meaning given in the formation of Jk) and suppose
that B^h^Bk (note that B^\). Thus A^m and so (m\)h\an ((m!)2|a„ by our
construction). Hence anh~x em\stf(m). Since
X Bk^ h^B kX>
this gives anh~x efk as required to complete the proof. □
Remark It is not true that one could find a set £ as above with measure zero,
since if
a${Jn-\£ + m),
m,n
which would then be a set of measure zero,
Z X(«a) = 0.
n=l
5.4 UNIFORM DISTRIBUTION IN HIGHER DIMENSIONS
Write Uk for the unit cube [0, l)k in k dimensions, and define the discrepancy of
a sequence x„ by
£>„(*„)= sup
higZ n=l
(5.4.1)
Here ^ denotes a box with sides parallel to the coordinate axes and A(-) now
denotes A>dimensional Lebesgue measure. There are many other ways of defining
152 Uniform distribution
discrepancy now; for example, one could consider tilted rectangles or balls, or even
more generally one could take the supremum in (5.4.1) over all convex sets to get
what has been called the isotropic discrepancy [282]. We shall return to this
question later in this section when we shall discover that altering the sets allowed
for ^ greatly affects the discrepancy. If DN(xn)=o(N), then we say that xn is
uniformly distributed. We shall investigate DN(xn) by generalizing Lemma 5.1 and
so producing a £>dimensional analogue of Theorem 5.5.
Lemma 5.8 Let xn be a sequence in Uk and an a sequence of non-negative reals.
Let 0b be a box in Uk with sides parallel to the coordinate axes, and let N and L
be positive integers. Then there are reals 9l9 Q2, 63 with \6j\ ^ 1 such that
n n / a ml i i\ a ofc+i
X„€# + Z*
N
E 0„e(iwxn)
n=l
(5.4.2)
where \m\ =msix1^j^k\mj\ (w = (m1, ...,mk)), and
«m)= E f\(-^l+mm*((nmj)-\pjA-pj)\ (5.4.3)
rtij = 0 for j > r
where mk+1 = 0, the side lengths of @t are Pl9..., /?fc, and when mj = 0,
.-1* 1 *yjPj «J*r
min*((7rm,.)-\j3;, l-ft) = -
}-Pj *J = r.
Proof We first remark that the sum over r in (5.4.3) makes the result unusable
in certain contexts, but causes no difficulty in our present application. It is
obtaining the required lower bound that causes most problems in establishing
(5.4.2). In [124] we have presented an alternative approach which, although less
suitable here, has advantages in other areas. First we note that, without loss
of generality, we may assume that & has one corner at the origin. Let
y={yx, ...,yk)eMk, and write Xj(y) to denote the characteristic function of [0,/?,•)
extended to be periodic with period one. Also write
k
x(y)= II Xjiyj), Xj(y)=i-Xj(y)'
We then have
x(y)=i-Xi(yi)-Xi(yi)x2(y2) Xi(yi)x2(y2>--Xk-i(yk-i)Xk(yk)-
This representation allows us to obtain a lower bound for x(y) from upper bounds
for Xj(yj) and Xj(yj) (thus avoiding the problem that our lower-bound
approximations can be negative). We manufacture such upper bounds as in the proof of
Lemma 5.1 so that x(.y)^l— f(y), where
Ay)=fi(yi)+Myi)f2(y2)+-
Uniform distribution in higher dimensions
153
and XjiyXfj(y), Xj(yXfj(y) (note that/,- is not 1-/}). It is clear from our
earlier discussion that
fi(yi)~-fr-i(yr-i)Tr(yM-(m1y1 + — mryr))dyl — dyr
ii/»
r-\ n
n
J] /,(>0e(-m,.j;)d.y
;=i Jo
/r(.y)e(-mrj>)d.y
^ fit v+min^Twij)-1,^,!-^)J,
where min* is as defined above and v = (L + l)_1. We write
d(m) =
f(yM-my)dy.
w
We note that there can be a contribution from fiiyO •••fr(yr) to d(m) only if
ra, = 0 for ally>r, since otherwise there is a factor
m
e(—hy)dy
involved in the evaluation of the Fourier coefficient, where h^O. We therefore
have \d(m)\^c(m). Also
where
rf(0) = l-(l-j81 + v)-(P1 + v)(l-j52 + v)-
Since 7\ < [k(k+1)]/2 and
^<ZrCj.
r = 7
this establishes the lower bound required for (5.4.2). The upper bound needed to
complete the proof is easily obtained by considering Xi(^i) • • • /*( J^)- □
We are now in a position to generalize Theorem 5.5. The result obtained is
usually referred to as the Erdos-Turan-Koksma theorem (see [171] and [259],
where Sziisz proved it independently). The constants which we state here appear
to be better than any others stated previously.
154
Uniform distribution
Theorem 5.21 Let xn be a sequence in IRfc, and L, N positive integers with L ^ 4.
Then
n , x k(k + \)N 2k+1N „ _ /fx
Av(*nK „/r ,'x +7F-TT2 + Q I KA)
0<|A|<L
2(L + 1) " (L + l)2
Z e(Ajt„)
n=l
(5.4.4)
where
and
KAr^nmaxd,!/*;!)
Cfc = (7r + l)((l + l/7r)fc-l).
Remark On taking fc = l, and omitting the middle term of (5.4.4), we recover
Theorem 5.5.
Proof In view of Lemma 5.8, we need only show that, for A#0,
£ nfv + minfl^ll^C^A).
r=l J=l\
hj = Oiorj>r
n\hj\
(5.4.5)
Let us suppose that /zs#0,1 O^r, and / of the h3 are zero, with 1 ^/ <r. Then,
if hj=0 for all j > r, we have
fl (v+ 111) ^ A +WO +*)'**)
<(l + l/7r)rr(A).
On summing over r, this gives (5.4.5) as required.
Corollary Let a e IRfc. The sequence no. is uniformly distributed if and only if
1, al5..., ixk are linearly independent over Q.
Remark Since the set {aeIR*: l,al5...,ak linearly dependent over Q} has
£>dimensional measure zero, the corollary implies that net, is uniformly distributed
for almost all a.
Proof If {1,0^, ...,ak} forms a linearly independent set over Q, then, given L,
the expression
r(h)
o<w<i.llMi + -+Ajka*l
is finite. By Theorem 5.21 and the bound
I e(jfrz)
01
Uniform distribution in higher dimensions
155
we obtain
(5.4.6)
Now, given € > 0, we may pick L so large that the first two terms on the right-hand
side of (5.4.6) contribute ^1/2* N. Then, for all large N, CkS(L)<\/2eN. Thus
DN(na)<€, which establishes the result.
On the other hand, if {1,^,...,ak} forms a linearly dependent set over Q,
there exist integers nl9..., nk such that
wi«iH t"wfcafce^-
Hence w^wajH hwfc{wafc} eZ for all integers w^l. Since there are only finitely
many integers h such that the hyperplanes
nlpl+ — +njk=h
intersect Uk, there is a box 3b with sides parallel to the coordinate axes having
positive £>dimensional volume, yet wa£^ for all w^l. This completes the proof. □
To obtain a quantitative version of the above corollary, we need the following
result given by W.M. Schmidt [240].
Lemma 5.9 Let k increasing sequences of positive integers a^ri), l^i^k,be given.
Let 6 be an arbitrary real number, and put
L L / L
£(L;<*!,...,afc)= £ •" Z (tfi-'-tfJ Z a,-0«(0,-) + 0|
9i = l «*=1 \
-1
i=l
Then we have for €>0, and almost all ae Uk, that
X(Z,;a1,...,afc)«(logL)k+1+e.
Proof We write rj = e/(k + l\
(5.4.7)
z=
G<y) = G{y,n) = {y\\ogy\'+ri)
l+IJN-l
and note that
1/2
1
10 *7log"2
Put
J(<li>'--><lk) =
G(Z)da.
U*
156
Uniform distribution
The change of variables ft=afaf(0i) + 0 gives
«%!»• ••»&) =
1
«i(tfi)-••«*(?*)
o
r<M<?*)
Id^.-.dA
17*
I ft
n
Id^.-.d^
<WII)dft
where we have used the periodicity of || x ||. Hence J(q 1,...,qk) = 2(?y(log l)*) 1; in
particular, the expression is finite and independent of ql9...,qk. It follows that
oo
oo
Z ••• Z ^i)---^*)^(?i>-••>?*)
$1 = 1 4*=1
is convergent, and so
00
00
X - I G(q1).--Giqk)G(l)
converges for almost all aeIR*.
By Lemma 1.2 the inequality
(5.4.8)
X^fai •••?*)
-2
(5.4.9)
has only a finite number of solutions in integers ql9... 9qk for almost all aeIR*.
The proof of the lemma will be completed by showing that (5.4.7) holds for the
set of almost all a, say sf9 for which both (5.4.8) and (5.4.9) hold. For ctestf we
have X^C_1(<7i • <7k)~2 for some C=C(a). Thus
|logS<21ogfa1...fc) = logC.
We thus have
oo
oo
□
J](L;a1,...,aJkKJI/£ ...£ G^)... Giqk)GC£)
<(logL)(1 +')*aogL*)1 +"«(logL)*+J +€
as desired to complete the proof.
Combining (5.4.6) with (5.4.7) gives the following result (after splitting the
summation over h into 2k blocks according to the sign of hi9 and noting that if
almost all a e s/9 then, for almost all a, (ax( — l)bl,..., afc( — \)bk) e $£ for all choices
of bje {0,1}).
Theorem 5.22 Let e > 0. Then for almost all aeUk we have
DN(na)«(\ogN)k+1+€. (5.4.10)
Remark Since Schmidt also demonstrated that
I(L;a1,...,ak)»(logL)fc+1
Uniform distribution in higher dimensions 157
for almost all a, we see that (5.4.10) is essentially the best result that can be
obtained using the Erdos-Turan-Koksma theorem. It is known that any sequence
xne Uk has £>N(jtn)»(logA0fc/2 for all N [224]. From this it follows, by considering
the sequence (xlH,...,xkn,nJN) in IR*+1, that DN(xn)»(\ogN)(k+1)/2 infinitely
often. Beck and Chen [26] and Pollington [224] have given conflicting conjectures
on the correct order for these functions. Beck [25] has demonstrated that if \j/ is
any non-decreasing function, then
DN (« a)«(log N)k iKlog log N)
for almost all ae IRfc, if and only if
oo
£ *l> \n)<co.
n=l
Beck's starting point is the application of the Poisson summation formula to
obtain
N
DN(ct,x):= X Z l-xl...xkN
meZk n=1
1 —Q(jnj Xj)\ 1 — e(A^(m1 a1 + —h mk ak))
2%m} ) 2n(mla1+—\-mkak—mk+lY
which he describes as 'a sum of almost pairwise independent functions of x\ The
problem reduces to a very careful consideration of when the denominator is small,
which brings one back to an analysis of ||m1a1H \-mk<xk\\. The methods of
Schmidt ([236] and [239]) are modified to complete the proof.
We leave it to the reader to verify that Theorem 5.11 generalizes as follows.
The result corresponding to Theorem 5.13 (that is, with the log power reduced
by 1) has been given by R. Nair [205]).
Theorem 5.23 Let e > 0, and suppose an to be a sequence of distinct integers. Then,
for almost all a e Mk,
DN(aLan)«Nll2(\ogN)k+3l2+*.
We conclude this chapter with a discussion of the discrepancy function
involving boxes in arbitrary position, which we shall denote by D%(xn). The fact
that the boxes may now be 'tilted' greatly alters the discrepancy. We have ([23],
[194], [244])
DmN{xH)»N«k\ with <7(k)=iQ-l/k),
while sequences exist with D*N(xn)«Na{k)+e. It might at first seem reasonable to
suppose that sequences wa would be amongst those of least discrepancy. This turns
158 Uniform distribution
out not to be the case. It is very easy to see that for A: = 2 we have
D*N(noL)^N113
for all N. For, by Dirichlet's theorem, there is an integer h with
HfoxJ^Ar173, ||/*a2KAT1/3, h^N2'3.
It follows that the points ({m/za1},{m^a2}), m^N113, all lie on a single ray from
one corner of [0, l)2. Hence there is a tilted rectangle with arbitrarily small
breadth, and length at least one, containing JV1/3 points.
To obtain the correct answer for the above problem we should instead apply
Dirichlet's theorem to \\ha1+ha2\\, which will show, in the general case, that
points net. lie quite close to certain parallel hyperplanes. We can then find a box
'between' the hyperplanes containing no points net. This is similar to the argument
we used to show that if {l,a1}..., ak} is a linearly dependent set, then net is not
uniformly distributed. The reasoning just outlined leads to the correct order of
discrepancy for net, as we shall subsequently demonstrate.
Theorem 5.24A For every eteUk and all N, we have
D*N(naL)»N1-1/k. (5.4.11)
Here the implied constant depends only on k.
Remark We note that the exponent in (5.4.11) is 2c{k), that is, the sequence net
has discrepancy at least the square of the minimum discrepancy.
Proof Given N, we can find hx,...,hk with
P1a1+-+/zfcafcK(4A0-1, \hj\^(4N)1/k.
Thus, for any integer n^N, there exists an integer m such that
i^||«(A1a1+---+Akafc)|| = |«(/z1a1+---+/zfcafc)-m|.
It follows that the regions
J <!/*!*!+•••+/zfcxfc-m|^J
contain no points net—m, iweZ*. Since the intersection with Uk of some of these
regions contains boxes with volume »N~1/fc, this proves the result.
We note that the above proof gives boxes which are 'thin' in one dimension,
but 'fat' in all others. Beck's lower-bound discrepancy result was demonstrated
using only hypercubes. We therefore note in the following that with a little more
work, we too can pick hypercubes with large discrepancy.
Theorem 5.24B Write DfN(xn) for the discrepancy function with respect to
hypercubes in Uk. Then, for every ete IR\ and for all N, we have
D]i(nct)»N1-ll\
where the implied constant depends only on k.
Uniform distribution in higher dimensions 159
Proof Pick two hyperplanes h1x1 -\ \-hkxk—m which intersect Uk and which
are a distance 1/4 +0(e) apart, where € is the distance between the hyperplanes
h1x1 + —\-hkxk = l/4 and hyXil \-hkxk = \ll. We may assume that there are
many hyperplanes intersecting Uk, for otherwise it will be easier to find a large
hypercube containing no points. We may assume e<N~k~\ for otherwise there is
a hypercube with side length € (and so volume »N~1/k) containing no points. To
simplify the argument, we shall suppose that the two hyperplanes picked are
exactly 1/4+ € apart. Now consider a hypercube Hl with one face on one of
the chosen hyperplanes, say 3^lt the opposite face a distance € from the other
hyperplane. Then Hx has all sides of length 1/4. Consider also a hypercube Cx
contained in H1 with one face on 34? l9 sides of length 1/4—e, and sharing one corner
on 34?x with Hv Suppose the discrepancies of Hx and Cx are both less than R. Let
the hyperplane containing the face of Cx parallel to (but not on) 3^1 be <2f2> and
the hyperplane containing the face of Hx parallel to but not on 3? x be 34? 3. There
are no points wot—m in that part of Hx lying between 3^2 and 3tifz. Hence the region
Vl consisting of that part of H1\C1 lying between jf1 and 34?2 contains
>N((i)k-(i-e)k)-2R = keN(i)k-1 +0(e2N)
points (note that the expected number is (1/4)*"* eTV. We can repeat this argument
with hypercubes H2, C2, //3, C3,... after picking a direction perpendicular to one
of the faces of Hl (not those parallel to 34? J, and moving the common corner of
Hp Cj along € each time in this direction. Let ra = [(4c)~ *]. We note that C, <= Hm
for l^y^m. Assuming that each //, and C, has discrepancy at most R, we
therefore obtain
(i)kN+R^m(N((i)k-(i-€)k)-2R)
=Nk(i)k-R(2ty1 + 0(eN).
Hence
R^(i)kN(k-\)e + 0(e2N).
Since N~k~2»£»N~k~\ this completes the proof. □
Theorem 5.25 For almost all a e Rk we have
D*N(na)« tf1-*",(logtf)1+**,+<. (5.4.12)
Proof First we need the Erdos-Turan theorem corresponding to this situation,
namely
N
DnM « T + SUP S c(w)
*- *<»> me 2*
0<|m|<L
N
Z e(f«*n)
n=l
(5.4.13)
160 Uniform distribution
Here
c{m) = fl mini 1,, . t M ), Lr(m) = £ a m•+mr,
r=l V |£r(«)l/ J-i
and the supremum is taken over all c(m) such that the hyperplanes
Lr(x) = 0, r=l, ...,&,
are mutually perpendicular. For the sequence net. there is no significant loss in
taking the supremum inside the summation in (5.4.13). We then note that if
\m\^L, then
|c(w)|^ min \mj
-i
<L*-innm(l,rV).
/.A V \mj\J
The proof then follows as for Theorem 5.22 upon taking
L=Arfc-1(logA0"(1+*",). □
We remark that it is easy to connect what is happening in the geometric
proof of Theorem 5.24 and the analytic proof of Theorem 5.25, with both
hinging upon the consideration of ||m1a1H \-mkak\\. In Theorem 5.25 we note
that c(m) might only be \tn\~1 in size, while \\m1<xl-\ \-mkock\\ could be \m\~k
in size. This is where the situation differs greatly from boxes with sides parallel
to the coordinate axes. In this case the characteristic functions have Fourier
coefficients which factor out as a product of terms |m1|"1|m2|~1... |mfc|_1. We
also note the difference with the situation covered in Chapter 3, where a box or
even a sequence of boxes & is given and then nae@k + m has solutions even for
quite small boxes.
One can obtain a result slightly weaker than Theorem 5.25 by noting that
for any sequence xn (see Theorem 1.6 in Chapter 2 of [175]). This can be proved
by considering approximations to a box by unions of cubes with sides aligned with
the axes having side length (DN(xn)/N)k'\ and the result holds for discrepancy
defined with respect to convex bodies.
It is possible to study discrepancy with respect to balls instead of boxes. Here
the minimum discrepancy is the same as that for boxes in arbitrary position (see
[26] Chapter 6, or the discussion in [194] or [195]). However, the discrepancy of
wa is now «Ni(k)+\ where £(&) = 1—2/(& + l). The author conjectures that this is
the correct order of size, but has not been able to find a proof of the corresponding
lower bound.
Notes
161
NOTES
For a fuller treatment of the subject of uniform distribution we refer the reader
to the works of Kuipers and Niederreiter [175] or Hlawka [137]. The
complementary topic of irregularities of distribution has been covered by Schmidt [244]
and Beck and Chen [26]. As mentioned in the Introduction, the subject has its
origins in problems to do with applied mathematics. Now it can be seen as a
fundamental component of analytic number theory impinging on many other
aspects of the subject. The connection between the discrepancy of the Farey
sequence and the Riemann hypothesis is well known [98], but many problems
in analytic number theory involve the estimation of exponential sums, and here
there is often an interplay with uniform distribution (see [142] for example).
Nevertheless, connections with numerical integration and other areas where
well-distributed random sequences are needed make this still one of the more
applicable areas of number theory. See [206] for a collection of many results in
this area. For applications it is often not enough just to have xn uniformly
distributed with small discrepancy. It is necessary to have the sequences
(xn,xn+1,...,xn+k) uniformly distributed in [0,l)fc for every k (see [99], [162],
[207], and [208]). It was shown by Koksma [163] that a" satisfies this criterion
for almost all a>l. See also the work of Jagerman [145]. For another problem on
the connection between sums and integrals, see [11].
Our exposition has not followed historical lines here, since we left one of
the important original results (the Weyl criterion [280]) until after the Erdos-
Turan theorem [96]. Ruzsa [233] has shown that the Erdos-Turan inequality is
essentially best possible for a wider class of examples than the one we supplied.
Since {ocbn} being uniformly distributed modulo one is equivalent to a
being normal to base b for b an integer exceeding 1, it is natural to define
normality to a non-integer base /? by the requirement that {a/?n} be uniformly
distributed modulo 1 (see [198] for example). By the results of this chapter,
almost all a are normal to any given base j5>l. We shall discuss this further in
Chapter 10.
For a detailed study of the discrepancy of the sequence wa, see [247]. For an
alternative approach to Theorem 5.21, see the work of Cochrane [63].
The results in Section 5.4 on the discrepancy of wa with respect to tilted
rectangles as stated appear to be new, but Zaremba [281] has given Theorem
5.24A for the special case a=g/p, where g e Z*. We remark that the generalization
to ^-dimensions of Koksma's inequality (due to Hlawka [135]) involves a
definition of bounded variation (due to Hardy [109] and Krause [173]) which
excludes functions such as the characteristic function of a tilted box. This is not
surprising if one is using the discrepancy function with respect to boxes aligned
with the coordinate axes, since we have seen that the discrepancy with respect to
tilted boxes can be much larger. Hlawka gave an alternative generalization [136],
which we now describe. Let 0> be a partition of [0, l)fc and write s(/;^) for the
difference between the upper and lower Darboux sums for / corresponding to &.
162
Uniform distribution
Let n{@>) be the maximum length of all the edges of the boxes in &. For a real
number a, set
S(/;a)= sup s{f;0>)
n(.&)*Za
Then Hlawka's result is that
E{f,N,xH) =
1 N
n — i
yu)d*
u*
«S(/;<5),
with
<5 = (Ar-1Z)N(jcJ)A
It is possible to generalize the variation we wrote as V*(f) to several dimensions
by writing
*{f9a) = {xe[09lf:fix)>a}9
and writing K(f9oc) for the number of convex sets required to make up <£"(/, a)
(in terms of unions and differences of sets). It is then easy to show that
E(f,N9xm)^jjV*(f)DN{«9xu),
where DN(<£,xn) is the discrepancy with respect to convex sets, and, as before,
n/)=
K(f9 a) da.
Zaremba [282] gave a result of this form for an integral over a convex subset of Uk9
but he still required the Hardy-Krause variation in place of our V*(f). The
measure V*(f) has many advantages over the Hardy-Krause definition. For
example, it assigns the value n to the characteristic function of a set of n convex
bodies in [0,1)\ or to many approximations to such a function. Indeed, it has the
same isotropic property as DN(^,xn).
For generalizations of the concept of uniform distribution to compact spaces
or topological groups, see [175]. For a generalization of the uniform distribution
of a" to s x s matrices, see [185] (here almost all a > 1 becomes: almost all matrices
with greatest eigenvalue exceeding 1 in modulus). We also note that Larcher [176]
has proved a general metric theorem which gives both Theorem 4.5 and Lemma 5.9
as corollaries.
Notes 163
In 1973 Erdos suggested to R.C. Baker the following variant of Khintchine's
conjecture for investigation:
Let /(a) be a bounded measurable function with period 1. Is it true that
1 N 1
AMco lOgiV^H
/(a) da
o
/or almost all a?
So far as the author is aware, this question remains open. Marstrand's
construction does not give a counter-example in this situation.
6
Diophantine approximation with
restricted numerator and denominator
The convergence case. Failure of the zero-one law. General criteria.
Sequences of number-theoretic interest. A fundamental lemma for non-
periodic problems.
6.1 INTRODUCTION AND STATEMENT OF RESULTS
So far, most of our results in metric number theory have been periodic in a:
changing a to a + w (or a to <x+n in IRfc) has not altered the property under
discussion. The only exceptions were in the previous chapter. Once we restrict
numerator and denominator in Diophantine approximation problems, we lose
periodicity. For our present discussion, one difficulty this produces is that we
may no longer be able to establish a zero-one law (see below, where we
construct an example in which such a law fails). However, Lemmas 1.6 and 2.3
involved no appeal to periodicity, and we combine them in Lemma 6.1 below to
produce what will be the starting point for many of our proofs in this and the
subsequent two chapters (where further non-periodic problems are considered).
In some cases we are able to use asymptotic formulae known when only one
variable is restricted (Theorems 3.1, 4.2, 4.3) to obtain results, even producing
asymptotic formulae when two variables are restricted (see Theorems 6.4 and 6.5
below).
The questions we address here were first considered in [117], [118], [119], and
[120], and concern the number of solutions for almost all a > 0 to the inequality
|a«—m\<\J/(n), nestf, wel, (6.1.1)
where stf, 31 are infinite sets of positive integers, and ^(w)e[0,i). We may also
want to impose the condition (m,«) = l in (6.1.1). This problem is too general: we
shall need to impose certain restrictions to tackle it. Note that although at first
sight there appear to be three 'variables': s#, ^, \J/, the restriction of n to stf could
be included by a redefinition of \J/. As might be expected, it is easier to work with
a 'nice' function \j/ and consider the restriction of n to s# separately. We shall
Introduction and statement of results 165
often suppose that there is some continuous non-increasing function p(x)
describing the probability that n e ^, that is,
N N
£1= 2>(«)(1+0(1)). (6.1.2)
n=l n=l
neSt
For example, we have:
& p(n)
prime numbers (log n) ~x
square-frees din2
n = a(modq) \lq
n = r2+s2 K/(\ogn)1/2
>=3(mod4)\ F /
"(1/2)
where K is the known constant:
2-u/2>
p=3(mod4)
We shall also suppose that either
\J/(ri)/n is non-increasing (6.1.3)
or that there exist constants au <r2 with
0<ox<\l/{m)l\j/{n)<<j2 for«^m<2«, w^l. (6.1.4)
We begin by giving the necessary condition for there to be infinitely many
solutions to (6.1.1) for almost all a, before embarking on the much more difficult
task of finding various sufficient conditions. The problem of finding solutions to
(6.1.1) for any given irrational is extremely difficult, and only quite weak results
have been given to date (except in the case «^ = ^ = the set of square-free integers
[131]).
Theorem 6.1 In the notation above, if the sum
Z P(")<K") (6.1.5)
nes/
converges, then there are only finitely many solutions to (6.1.1) for almost all a > 0.
Proof For T > 0, write J = [T, T+1],
in|J ((m-\l/(n))/n,(m + *l'(n))/n) if nerf.
166 Diophantine approximation with restricted numerator and denominator
Then
n=l nessf n \ me& /
\m/n-T\^2
= S2^( I rfmXl-Ml)))
\m/n-T\^2
« Z *K")p(")
ne.s/
if either (6.1.3) or (6.1.4) hold. The result then follows by Lemma 1.2. □
Remark One immediate corollary of Theorem 6.1 is:
Almost all oc > 0 have only finitely many conver gents plq to their continued-fraction
expansion with both p and q prime.
This contrasts with the case when only one of p or q is restricted, as we saw in
Chapter 2.
We now know that the divergence of (6.1.5) is a necessary condition for (6.1.1)
to have infinitely many solutions for almost all a > 0. We shall prove that it is
also sufficient for the combinations of sets s#, & of greatest interest in number
theory. First though, we shall provide an example to show that we cannot assume
a zero-one law a priori. We do this by choosing sets $£, 38 so that the ratios m/n,
me@,ne<stf, are not even dense in [0,oo).
For S > 0, put
j^ = ((1 + <5)2"(1 + <52)", (l + <5)2n+1(l + ^2)n),
/„ = ((l + <5)2n+1(l + <52r, (l + <5)2«+2(l + <52y),
jTn = ((l + <5)2''+2(l + ^2)'', (l + <5)2n+2(l + <52)n+1),
^=u^> /= u /, .*■= u •*■-.
n=0 n=0 n=0
We note that J, #, X are pairwise disjoint. Now, if mel, nest, we have
n v
with
(l + <5)2r+1^w<(l + <5)2r+2and(l + <5)2s<i;^(l + <5)
Introduction and statement of results 167
for some r,s. Hence
(l + <5)2r-2s<-^(l + <5)2r-2s+2.
v
Thus mn~1eJr_s\jfr_s and so mn'1 $X. We have thus shown that the set of
fractions mn'1 is not dense in the set of positive reals. If we write
"-« Nn=l
neS
for the lower asymptotic density of a set of positive integers (as defined earlier in
Chapter 2), then clearly d_(st) = d_(@) > \ — 3<5. This shows that Theorem 6.6 below
is best possible. Of course, in this example there is no continuous non-increasing
function p(x) for which (6.1.2) holds.
We state two general results first before moving on to those sets of special
number-theoretic interest. We require some more notation:
^(A0=I1, ¥(AQ=2>("), <p(D,n)= £ 1.
n=1 n=1 m=1
ne.s/ nes/ (m,D)=l
(Note that (p(D, n) has the same meaning—and, indeed will play a similar role—as
in Chapter 4.) Henceforth we assume that *F(oo) diverges. Also, we shall have
cause to refer to the following conditions:
^^P->c + l foralliV>l (6.1.6)
A(N)
for some constants K>\, c>0, depending only on stf\
A(2N)-A(N)<C for all JV> 1, (6.1.7)
where C depends only on s/;
lim lim(ra,w) = oo; (6.1.8)
n-»oo m-»oo
nes/ me3
lim Hminf-J- £ ^^ = 1; (6.1.9)
nes/
liminf-^-X- I l>c(y-fi (6.1.10)
at-oo A(N) „=t n *-"•
meat
ne.s/ m/ne[fi,y]
(m, n) = 1
for all y > /? > 0, where c is a positive constant depending only on stf and ^. We
note that (6.1.6) and (6.1.7) are opposite extremes: the first says that s# is not too
168 Diophantine approximation with restricted numerator and denominator
sparse; the latter that s# is very sparse. Either condition is helpful in our proofs,
just as we have seen in earlier chapters.
Theorem 6.2 Let stf and $8 be sets of positive integers, where a\@) > 0. Suppose
that at least one of (6.1.6)—(6.1.8) holds for $£ and at least one of (6.1.3), (6.1.4)
holds for if/. Then, if (6.1.10) holds, there are infinitely many solutions to
\ocn—m\<\J/(n), nestf^ wel, (ra,w) = l (6.1.11)
for almost all a > 0.
Corollary Almost all positive a have infinitely many convergents mln to their
continued fraction expansion with any of the following conditions on m, n:
(i) m, n both square-free;
(ii) m = a (mod/?), n = b(modq), where (a,p,b,q) = l;
(iii) one of m, n prime, the other square-free;
(iv) one of m, n prime (or even a prime in an arithmetic progression), the other in
an arithmetic progression.
Proof of Corollary We note that dividing (6.1.11) by a enables us to swap the
roles of stf and 31 if and when necessary (in (iii) and (iv) when m is prime), so we
may assume a\@) >0 in the following. Clearly (6.1.6) holds for each of (i)—(iv),
while il/(ri) = V2n satisfies (6.1.3). The condition (6.1.10) is easily shown to hold.
We establish this for (iii) by way of illustration. Say n is prime and m is square-free.
Then
1 " 1 _ . 1 " 1/6
N 1 1 N 1 / f\ \
A(N)n%n £m A(N)
ne.s/ (m,n)=l ne.s/
m/ne[p9y]
nes/
y-P
2
for all large N. □
Remark For case (i) we note that certain members of the exceptional set can
easily be explicitly constructed. For example, [0;4,2,4,£] = (45 —101/2)/186 has
convergent denominators alternately divisible by 4 and 9. It is possible to prove
an asymptotic formula for all the cases given above (see [118] for the method of
proof for cases (iii) and (iv); see Theorem 6.5 for case (ii); an alteration to the
proof of Theorem 6.4 invoking Theorem 4.4 will establish case (i)).
Clearly, other corollaries are possible. For example, one could take s/ = {rm:
m = l,2,...}, where reZ, r ^2, and & is the set of square-free integers.
Alternatively one could put s£ = {/(«): «eN}, where / is a polynomial with integer
Introduction and statement of results 169
coefficients. We note that in the hypothesis of Theorem 6.2 the condition d_{@) > 0
is in fact implied by (6.1.10).
Theorem 6.3 Let the hypotheses of Theorem 6.2 be given, with (6.1.9) substituted
for (6.1.10), and 31 now must have positive density. Then there are infinitely many
solutions to (6.1.1) for almost all a > 0.
Remark This deals with all cases where a\stf) > 0 and 3 has positive density.
We now present some quantitative results.
Theorem 6.4 Let a\stf) > 0, and suppose that \(/ is non-increasing. Write
Jf(s4', a, N) for the number of solutions to
| aw—m | < \J/(n), ne$#,m square-free. (6.1.12)
Then for almost all cc > 0 we have
JTW, a, AQ _ 12
if-Too TO " 7?
lim \JJs ' = -^. (6.1.13)
Corollary If we take s# to be the set of square-free integers in the above, we
obtain
^(^,a,AO = d + 0(l))^ £ <K»)-
Theorem 6.5 Let d_($#) > 0, and suppose that at least one of (6.1.3), (6.1.4) holds.
Write J^(s/, a, q, a, /?, N) for the number of solutions to
|wa—w + /?| <(/<(«), nestf, n^N, m = a (modq). (6.1.14)
Then, for any given real /? and e > 0, for almost all a > 0, we have
jr(jtf,a,q,tx,p,N)=2-^ + OC¥il/2)+i(N)). (6.1.15)
Q
Corollary Given \f/ and /? as above, and integers a, b, c, d, the number of solutions
to
\noi—ra + /?| <il/(n), n^N, n = a (mod b), m = c(modd)
is, for almost all a, equal to
2 N // N \d/2) + €>
for any € > 0.
Remark If (a, b,c,d) = \, it is possible to modify the proof to impose the
condition (w,«) = l, and thus obtain a quantitative version of Theorem 6.2, Corollary,
case (ii).
170 Diophantine approximation with restricted numerator and denominator
Theorem 6.6 Let \j/ be non-increasing and let st, & be two infinite sets of positive
integers such that
d(s/) + d(a)>l.
Then there are infinitely many solutions to (6.1.1) for almost all a > 0.
Theorem 6.7 Let ij/ satisfy at least one of (6.1.3), (6.1.4), and write & for the set
of primes, and £f for the set of numbers properly represented as the sum of two
squares. Then there are infinitely many solutions to (6.1.1) for almost all a > 0, where
stf,@e {0>, Sf}, if and only if
„f2(log«)1+e
diverges, where
fO if jf = @ = y
9 = h if ^ = ^\ & = &' (or vice versa)
U if rf = @ = 0>.
Corollary Almost all positive a have infinitely many convergents to their continued-
fraction expansion with numerator and denominator simultaneously the sum of two
squares.
Remark The words 'almost all' here certainly cannot be replaced by 'all
irrational', since 3 +1/51/2 (= [3; 2,4]) has every convergent numerator congruent
to 3 (mod 4). The author has given in [125] a construction of numbers not well
approximable by fractions of the type in Theorem 6.7. For approximations to this
problem for all irrational a, see [144], [113], and [114],
Finally we mention a result which is a corollary to Theorem 7.2 in the next
chapter.
Theorem 6.8 Let /(«) be a polynomial of degree at least two with positive leading
coefficient (not necessarily having integer coefficients). Then there are infinitely
many solutions for almost all a > 0 to
I a/(«) -p\ < (n log log 9«)"J
and to
\xf(q)-p\ <(\ogq)(q\og\og9qr\
where p and q denote prime numbers.
Remark This deals with cases such as s£ as the set of squares and ^ = ^, and
is nearly best possible in the sense that there would only be finitely many solutions
for almost all a>0 if the inequalities had been sharpened by only a factor
(loglogfl)'', say. Since Theorem 7.2 will deal with ny for any y > 1, it is possible to
deduce the following result (which is probably of less interest) using
\<mr-mk\<C(r,k)nr(llk-1)\anr/k-m\.
Proof of Theorem 6.2 171
Let k,r>\ be positive integers. Then there are infinitely many solutions for almost
all a > 0 to
\<mr-mk\ <«r(1-(1/fc))-1(loglog9«)_1. (6.1.16)
Moreover, if k\r or r\k, and k^r, then the right-hand side above may be replaced
by \l/(ri)nr~r/k for any function \j/ such that the following sum diverges:
oo
2>(n)—-•
The final part of the above follows from the fact that the Duffin and Schaeffer
conjecture is true for an = nk (Theorem 3.7, (i)), and is best possible. The result
(6.1.16) is near to best possible, since the right-hand side cannot be reduced by a
factor (logh)-1 (k>glog")-'' for any 77 >0. To see this, note that the probability
that a number of size nr is an exact kth power is of order «"r(1 ~(1/fc)).
6.2 PROOF OF THEOREM 6.2
We begin with a lemma that will be fundamental to much of our work here and
in the subsequent two chapters.
Lemma 6.1 Let J be a subinterval of U, and Q)n a sequence of subsets of J. For
each open interval # <=,/, suppose that there is a sequence of sets l.c^n/ such
that
tw*) = «> (6.2.1)
n=l
and
limsupf X K&S) ( I K@nn<%m)) >31(f), (6.2.2)
where 5 is a positive constant independent of f. Then almost all cue J belong to
infinitely many Q)n.
Proof This follows by combining Lemmas 1.6 and 2.3. □
To prove Theorem 6.2 we shall apply Lemma 6.1 with
I ifm^jaf
^mr±m\ i(me^ (6-2.3)
rim \ m m J
(m 1*^=1
andput^m = ^mn/. v
Hence (6.2.1) follows from (6.1.10), the conditions on ^, and the divergence of
¥(00). If/ = (j5,y), we obtain
% Mm)>2c(y-py¥{N), (6.2.4)
172 Diophantine approximation with restricted numerator and denominator
for all sufficiently large N. It is at this point that we require 'liminf in (6.1.10)
rather than 'limsup'. It then suffices to show that, for some absolute constant C,
X WmnaH) < (C(y-p) + o(l))V2(N), (6.2.5)
since the desired conclusion then follows from Lemma 6.1.
To obtain (6.2.5) we note that, as in Chapter 2,
«*.n*.)<2»m(^,^)l II, (6.2.6)
V m n /.„,,,
\rn — sm\ < A
where ,4=max (m^(«),«i/r(m)), and we have changed our normal convention by
writing a~b for albe{fl,y). This altered notation continues throughout this
chapter. We shall henceforth suppose that (6.1.3) holds; the proof when (6.1.4)
holds is similar, but the constant in (6.2.5) will then depend on <jx and a2. We
also suppose, without loss of generality, that n^m, M^m< 2M, which enables
us to replace A by 2M\J/(ri) and take \l/(m)/m in the minimum. We then require
the following lemma to complete our estimation of (6.2.5).
Lemma 6.2 Let A>0, M^ 1, and n a positive integer be given, with n<2M. Let
$£ be a set of positive integers and (j5,y) an open subinterval of (0, oo). Then the
number of solutions to
\ms-nr\<A, (6.2.7)
with
r~m, 5~«, («, r) = (m,s) = l, m#«, mestf, M^m<2M
is
^AK(y-p) X 1 + OGE), (6.2.8)
me.s/
M^m<2M
where K is absolute, and
£ = U(2M)-A(M) if (6.1.8) holds, 2
\min(M,n(A(2M)—A(M))) otherwise.
Here, the constant implied by the O notation depends only on (/?, y), except when
(6.1.8) holds, when it also depends on the speed of growth of(m,ri).
We now complete the proof of Theorem 6.2 before proving Lemma 6.2.
Suppose firstly that (6.1.6) holds. Then, by (6.2.6) and (6.2.8), taking E=M from
Proof of Theorem 6.2 173
(6.2.9), for each fixed n we have
£ X{@mn@n)^K'(y-p) X <KmW(n) + o( £ ^Y (6.2.10)
mes/
The first term on the right of (6.2.10) when summed over nes/ gives the
C(y — p)^2(N) of (6.2.5), since by our conditions on \ji and sf we have
¥(2A0 < 2¥(JV). On the other hand,
I I *^< L^Um)«*(2tf),
n=ln^m<2iV W m^2N W
nes/
using partial summation. As we have already remarked, *F(2A0 ^ 2*F(iV), and
since *F(iV)-»oo, this establishes (6.2.5).
Now suppose that (6.1.7) holds. In this case we take E = n(A(2M) — A(M))
from (6.2.9). We then obtain (6.2.10), but now with an error term
» I 5^. (6.2.11)
n<m<2N m
me.s/
By (6.1.7),
m
n«m. (6.2.12)
n=l
nes/
Thus, summing (6.2.11) over nestf gives an error which is OC¥(N))9 as required.
When (6.1.8) holds, we may work as above, but the absence of the factor n in
E means that in place of (6.2.12) we can use the trivial bound A(m) ^ m. □
Proof of Lemma 6.2 We first note that the difference between this result and our
work in Chapter 2 is the restriction of r and s, which requires us to employ Fourier
analysis. The uniformity of the result in /?, y is, of course, vitally important.
Now write 6 = y — p. We may suppose that «_1 <6 <i, and A<n, for
otherwise the proof would follow instantly from our work in Chapter 2. Let h be
a divisor of n and consider the contribution of those m belonging to
stfh = {mestf:(m,ri) = h, M^m<2M}.
We note that the conditions m^n, («, s) = (m, r) = 1 ensure that there is no solution
with ms=rn, so we must have A^h for solutions to exist. Of course this
corresponds to how we proved Theorem 2.5. The number of solutions to (6.2.7)
is then no more than the number of solutions for v, s, a to
vs=a (mod#), \a\^A/h, tf#0, vhes$h, s~w, (6.2.13)
174 Diophantine approximation with restricted numerator and denominator
where g=nlh. If hd ^ 1, we can simply count the solutions in s for each pair v,a
to obtain a bound
1A
— (hd + \) X \^AA6 X 1. (6.2.14)
If /z0 < 1, write 5= [/4//z] and then we count solutions in u, 5, a of
us=0(mod0), |a|<22?, vhes/h, s~w, (6.2.15)
weighted with a factor v(a) = (2B—\a\)/B. Clearly this provides an upper bound
for the number of solutions to (6.2.13). The purpose of the factor v(a) should soon
become apparent: it will save an undesirable logarithmic factor when we estimate
some exponential sums. Now write n=—(p+y)h/2 and let v denote a solution x
to vx = 1 (mod g) (note that vh e stfh implies that (v, g) = l, while x will appear only
in functions having period g, so any solution would suffice). Solving (6.2.15) is
then equivalent to solving
va
Oh
2
<—, \a\^2B, vhes/h.
We now apply Lemma 5.1 with L = [(Oh) 1 ] to show that the number of solutions
to (6.2.15) weighted with the factor v(a) is no more than
26h X v(a) I l + o( X he^d^rv/g)), (6.2.16)
\a\^2B mes/h \vhesith r=l /
where d(B,x)=mm(B,B~1\\x\\~2). To obtain (6.2.16) we have noted that, for
11*11 #0,
v^ .... 1 fsm2nBx\2 AC.yn x
£ v(a)e(ax)=-(—. ^4<5(£,x).
m<2£ B\ smnx J
Now the main term in (6.2.16) is
%AB £ 1.
mes/h
To bound the second term in (6.2.16) we note that its value is not decreased by
changing the summation to include all v with (y, g) = \ and M^vh< 2M (this
summation condition will be tacitly assumed in the following). Swapping the
summation order then gives
Oh^^S^v/g). (6.2.17)
r=l v
Proof of Theorem 6.3 175
From nB > 1 we deduce that L < g, so that (r, g) < g. Thus, if 1 ^ r < L, for any K,
we have
V + ff-l <?/(r,s)
W=F 17=1
Hence (6.2.17) is bounded above by
8g(M/(hg) + 1)L0/* ^ 24M//z,
since gh=n<2M.
A different bound for the second term of (6.2.16) may be obtained by noting
that S(B, vr/g) ^ g, and so the sum is bounded above by
\ I 1. (6.2.18)
Assembling all our results so far then gives the following upper estimate for the
number of solutions to (6.2.7):
£M9 I l+of I min(j| I if)).
h\n me.s/h \ h\n \n m<=s/k "//
hd<l
Since
X X l = A(2M)-A(M)&nd £ lo'1,
h\n mest?h h6<l'1
this establishes (6.2.8) with (6.2.9) except in the case when (6.1.8) holds. To deal
with this case, we note that if (6.1.8) is valid, then, given 0, there is a number R
such that (w,«) ^ 0_1 for m,n^R. If we take the term (6.2.18) when n < R and
note that is/fc=0for h<B~1 when n^R, we obtain a bound from (6.2.14) and
(6.2.16) of
8A6 £ \ + 0(R(A(2M)-A(M))),
me.s/k
as required. (R depends only on 9 and the speed of growth of (m,«), to give the
result claimed.) □
6.3 PROOF OF THEOREM 6.3
The outline of the proof is the same as for Theorem 6.2. we write
0 if m$s/
U ((r—<A(w))/m, (r + (m))/ra) if me$4.
rei
Now we must be careful in our choice of the sets &„ and avail ourselves of
m
Schmidt's technique as described in Chapter 4. Let /" = (/?, y) <= (0, oo), and
176 Diopkantine approximation with restricted numerator and denominator
suppose that X(f)<\ (for otherwise the proof could be simplified). Suppose that
& has density c. Then, since (6.1.9) holds, we can pick D so that
n^oo A(N) „=! n 2
with d=y — ft as above. Hence
Hm » £^ I 1>^. (6.3.1)
(m,n)<£>
We put
if m$s/
rYa V m m J
(m,r)*ZD
r~m
Thus, by (6.3.1), we have
We must now obtain a satisfactory estimate for
X A(^mn^„). (6.3.2)
We may work as in the previous section, replacing (r, m) = («, 5) = 1 with (r, m) ^ Z),
(n,s)^D. This only brings in certain solutions of (6.2.7) with ms=nr, since we
used the coprimeness condition only to show that no such solutions existed. We
must therefore deal with an additional term
£ I%IL (6.3.3)
n=l n<m^N r^m s^n
nes/ mes/ re& s^#
rn = sm
(r9m),(s,n)^D
Now, if we fix a pair m, n and consider rn=sm subject to (r, m), (5, n) ^ 2), then
it is clear that unless (m, n) ^ raAD, there will be no solutions. When solutions do
exist, there are no more than In of them. Hence (6.3.3) is bounded above by
2£iKm) ^ n^2D2V(N),
m=l ™ m^n^m/D
mes/ (myn)^m/D
Proof of Theorem 6.4 111
since
£ n^ Y md^D
2m.
m^n^m/D d\m
(m,n)2>m/D d^D
It follows that the sum (6.3.2) is bounded above by
KdV2(N) + OmN))
for some absolute constant K, and Theorem 6.3 then follows from Lemma 6.1. It
should be noted that, as in Chapter 4, the parameter D is vital to exclude from Q)m
certain intervals which would have made X(@mr\<%n) too large on average. □
6.4 PROOF OF THEOREM 6.4
Let e and T be fixed positive quantities. We shall show that
,^G<a,A0 12
T^T TO > n
and
liminf \TI/\T[ ' > — -€ (6.4.1)
limsup —1TI/X^— < -ji + € (6.4.2)
for almost all ae [T,2T). Since € was arbitrary, (6.4.1) and (6.4.2) give (6.1.13). It
is possible to establish an asymptotic formula directly with an explicit (although
rather weak) error term, but this requires considerably more work. We shall be
using Schmidt's method again implicitly, for we shall appeal to Theorem 4.2.
Now, let y ^ 2 be a parameter to be chosen later, in terms of € and sd only,
and let
Y=Y[p, <% = {nd2:(d,Y) = \, \i\n) = \, d>y).
Also write ^K(a, N) for the number of solutions to
\ncL—m\<\l/{ri), me&, n^N,
put 5 = d(s#), and write
LC< a, TV, >;) = £>(<*) X 1- (6-4-3)
d\Y n=l
ness/
\an-md2\<\l/(n)
We note that
d\Y neS
n=0{modd2)
178 Diophantine approximation with restricted numerator and denominator
equals the number of members of £ which have no square factors dividing Y2.
Also, & consists of all non-square-free positive integers with no square factor
dividing Y2. Thus
0 < L (s/, a, N, y) - JT{^t a, N) ^ ^(a, N). (6.4.4)
Now, for each fixed d, the number of solutions to
\n(dd2)-m\<\l/(n)ld2, nejtf, n^N
can be estimated by Theorem 4.2 for almost all a as
?^ + 0(¥*(AO(1/2)+€),
dL
with
¥*(A/)= J>(").
Since *F(JV) ^i<5*F*(A0 for all large N, we replace *F* by *F in the above error
term. Hence, since Y is fixed (not depending on N), we obtain
L(jtf,a,N,y) = 2V(N)YJ^- + O(W0(1/2)+€). (6.4.5)
d\Y "
We note that
6 „ n{d) 6 2 fC Ar^
7T2 ^ rf2 n2 y
Thus, provided that j>>4€_1, (6.4.2) follows from (6.4.5).
Clearly, Jf(a, N) is bounded above by the number of solutions to
\m/<x-n\<d(m), mel, m^2TN+l,
where
[r1 for m<27+l
~ m(rn- \)IQT))I T for m ^ 27+1.
By partial summation,
X 0(m)<
m^2TN+l
for all sufficiently large N. It follows by Theorem 4.2 (considering almost all
ye[(2r)-1, r_1], and putting a=y_1) that, for almost all ae[r,2r], we have
4*FCAT)
^(a,A0 <—— + 0(¥(JV)(1/2)+€). (6.4.7)
Proof of Theorems 6.5 and 6.6 179
Combining (6.4.5), (6.4.6) and (6.4.7) gives, for almost all a,
jr(^,a,N)>"¥(N)(\2n-2-S(5yy1) + OC¥(N)ai2)+€).
Taking € = 8(<5y)_1 then gives (6.4.1) and completes the proof. □
6.5 PROOF OF THEOREMS 6.5 AND 6.6
By Theorem 4.2 (modified to consider ||«a + /?||), the number of solutions to
\\yn-(P-oi)/q\\<\l/(ri)/q, nest, n^N
is
^^ + O(¥(A0(1/2)+€) (6.5.1)
a
for almost all y. On setting oc = yq, the inequality becomes
\n<xJq + {f}-oi)lq-v\<\l/(ri)lq, n^N,ne&f. (6.5.2)
The proof of Theorem 6.5 is completed by noting that (6.5.2) gives (6.1.14) with
m = q(v-\-a/q)=a(modq). □
To prove Theorem 6.6, let a\s#) + dij%) = 1 + b. We consider ae[y,Y), where
y>0, Y=y + n, rj=yS/3. The number of solutions to (6.1.1) is then at least the
number of solutions to
\n—m/a|<^*(m)7_1, wel, m^Ny,
less the number of solutions to
\noc —m\<\j/(ri), n$#/, n^N,
where
\l/*{m) = \l/(\ + [mly]).
With two applications of Theorem 4.2 this gives, for almost all a > 0, the lower
bound with main terms
1 X iA*(m)- £ <Kn) = (i X **(m)+ £ ^(n)J- X ^(n). (6.5.3)
■* m^Ny n=l \* m^Ny n=l / n=l
mef n$.qf me& ne.s/
Since \J/(ri) is non-increasing, we can use partial summation together with the
definition of a\) to obtain
m ^ Aty n = 1
180 Diophantine approximation with restricted numerator and denominator
and
£ Wn)^(j/)(l + o(l))2>(")-
n=l n=l
Hence the expression in (6.5.3) is
N
>
(d(rf) + d(m-l-i5+o(l)) X MO-
n=l
Since d($#) + d{@l) — \—\5=\5>0i and the error terms from Theorem 4.2 are
smaller than the main terms, this completes the proof of Theorem 6.6. □
6.6 PROOF OF THEOREM 6.7
We shall prove the theorem only in the case <$# = & = &. The proof of the main
result required (Lemma 6.3) can easily be modified to deal with s/ = @ = Sf
(replace the three-dimensional sieve by a sieve of dimension 3/2). A judicious
appeal to Cauchy's inequality settles the case s/ = 0>, @ = Sf and vice versa. In
the following, /?, q, r, s will represent primes. We shall use Lemma 6.1 with
0 if m is composite
U ((s-ij/(p))/Ms+il/(p))/p) if m=p,
and &m = @mnf. For the sake of clarity we shall take f = (\,\ + 6). We write
a ~ A to mean A < a < (1 + 9)A, and b: B to mean B < b < (1 + 0)2B. The reason for
the latter notation is so that conditions like r~q, q~Q can be replaced by q~Q,
r:Q when obtaining upper bounds. We shall assume that 6<\, since the
important point, as we have seen, is keeping uniformity in 6 for the main term in
our overlap estimates as #-►().
Working as previously, we have, for p<q,
Uapnaq)^?^ XX 1. (6.6.1)
r~qs~p
0<\rp-sq\<2qil/(p)
Our aim is to demonstrate that
£ X(<%pn<%q)^C9V(N)2+HN, (6.6.2)
2^p<q^N
where C is absolute, and HN=o(V2(N)), with
Proof of Theorem 6.7 181
We split up the ranges of summation over p and q on the left of (6.6.2) into
subranges of the form p~P, q~Q. In view of (6.6.1), the sum of interest to us is
bounded by
I5(P,0, (6.6.3)
with P, Q taking values 2(1 + 0)" up to N(l + 6), P^Q, and
S(P,Q)=^Q IE 1. (6.6.4)
& r.Q ?.P
q~Qp~P
0<\rp-sq\<3Q*HP)
We now state two lemmas which will establish an adequate bound for S(P,Q)
and thus (6.6.3). The first of these will require some work on our part; the second
is a standard result from sieve theory. We shall give the proof of the following
lemma after we have completed the proof of Theorem 6.7.
Lemma 6.3 Let positive real numbers A, P, Q be given, with Q^P^6~2 and
AP>Q3/4. Then the number of solutions to
0<\rp-sq\<A, p~P, q~Q, r.Q, s:P (6.6.5)
is
^WFWQ + O{APQim0)' (666)
where C is absolute.
Remark No great importance should be attached to the exponents 3/4 and 19/20
above. The important point is that the result is of the correct order of magnitude
in P and Q and behaves correctly as 0->O.
Lemma 6.4 Let n(x) denote the number of primes not exceeding x. Then, for any
real numbers X and Y with 1 <Y^X, we have
n(X)-n(X-Y)<-^-. (6.6.7)
logr
Proof This is a simple form of the well-known Brun-Titchmarsh inequality. For
example, see Theorem 3.7 in [108]. □
Proof of Theorem 6.7 The philosophy of the proof is to use Lemma 6.4 when
Lemma 6.3 is inapplicable. Although this gives the 'wrong' dependence on 9, as
we shall soon see, the ranges of P and Q involved are such that these terms
contribute only to the error term HN in (6.6.2).
A 'trivial' bound for the number of solutions to (6.6.5) may be obtained by
considering rp—sq = h for each fixed h with \h\<A. This has no more than two
solutions in s, q when r, p, and h are fixed. This gives a bound from (6.6.7),
182 Diophantine approximation with restricted numerator and denominator
assuming that P^6~2, no more than
CAPQO2
log/> log 0
If Q\j/(P)^QZI*IP, then (6.6.8) gives
(6.6.8)
5(P0«^MW^ G3/V(0
Q logPlogg logPlogg'
The contribution of such terms to (6.6.3) is thus
^Qmil/(Q) log logO
which is easily absorbed into i/N.
For the terms in (6.6.4) with Q3/*/P<Q\(/(P) we substitute (6.6.6) to give a
total contribution no more than
I ^tp^y +o(YdPQ1"2(>HPmQ))<Cev\M)+o(V2(N)),
as required.
It only remains to consider the cases (i) Q^9~2, (ii) P^9~2<Q. The
contribution from case (i) is trivially 0(1) (there is no need to restrict the variables
to be primes in this case). For case (ii) we note that trivially,
2P2MQ) ^ P2Q)lt(Q)
Q tQ WQ
q~Q
by Lemma 6.4. Summing over P and Q thus gives a contribution which is 0(V(N))
to (6.6.2). This completes the proof of Theorem 6.7. □
Before we start to prove Lemma 6.3, we need to state two other important
results from analytic number theory.
Lemma 6.5 For square-free integers d, write
0*0-0(3-!)- 1 **.
p\d\ PJ lmn = d n
l,m,n>0
Let s$be a set of positive integers, Y, T positive reals, and h an even positive integer
with h^2Y20. Put
nest a
d\n
Proof of Theorem 6.7 183
Then the number of members of $4 having all their prime factors either dividing h
or greater than Y is
CT I h \
Y+tf( I fi2(d)d<\Rd\\, (6.6.9)
iog3rv<pW
(d,h)=l
where €>0, and the implied constant depends only on e.
Proof This is a form of the three-dimensional sieve, and follows, for example,
from Theorem 5.2 in [108]. □
In the following we write v for a solution to vx = \ (modvv), where (v, w) = l
and w will be either the modulus of a congruence or the denominator in a fraction
whose value need only be determined modulo one.
Lemma 6.6 Let U, V^l be real, and t, w be integers with (t, w) = 1, V<3w. Then,
for any € > 0, we have
i
VUt
«Uw(1/2)+e. (6.6.10)
0>,w)=l
Proof See Lemmas 6 and 7 in Chapter 2 of [138]. □
We note that the proof uses the deep methods of Weil for estimating complete
Kloosterman sums. We remark that the distribution of fractions on the real line
is intimately connected with the study of the equation ab—cd = \, and so this
appeal to results on Kloosterman sums should not be viewed as unnatural. Indeed,
various authors have discussed relations between continued fractions and the
study of the modular group (for example see [250]).
Proof of Lemma 6.3 We first note that the result is trivial if A > PQIl. We then
treat the cases P<Q1/4, P^Q114 separately.
Case 1: P<Q1/4. The number of solutions to (6.6.5) is no more than
6A
S.P S.P
r.Q r.Q
by Lemma 6.4
since A>Q3/VP
24A_ _1
<logepV '
s:P
CAPQ63
<log2giog2P'
184 Diophantine approximation with restricted numerator and denominator
using Lemma 6.4 again since PO^P112, Q9^Q1/2. This gives (6.6.6) in this case,
as required.
Case 2: P^Q114. Here we wish to bound the number of solutions to
rp—sq=h
with 0<\h\^A. Without loss of generality we shall treat h>0 only. The
contribution from those h with p\h is trivially «AQ«APQ19'20, and is thus
absorbed into the second term in (6.6.6). We note the fact that h #0 was vital here.
For other values of h we apply Lemma 6.5 with
s0 = {Imn:lp—mn=h, m:P, w~0,
and Y as some suitable value not exceeding P. We remark that we have lost
nothing by not using any averaging over h, since the crucial case has P near to Q
and A 'small' (less than any power of Q). If A ^ Q8 for some S > 0, then the analysis
of the remainder term Rd is elementary (making use of averaging over h). This
would suffice if \J/(q)>q'~1 for some e>0, but not in the 'best possible' situation
of our theorem (where one might take ij/(q) = q~* log#, for example). It is for this
reason that we appeal to the 'deeper' results implicit in Lemma 6.6.
Clearly the number of solutions to
wejs/, « = 0 (modJ),
for square-free d, is equal to the number of solutions to
xyz=d, x,y,z^\, mnxy=—h(modpz), m:P/x, n~Qly. (6.6.11)
Now, if (d,hp) = \, then (/z,/?z) = l, so (xymn,pz) = \. We may thus rewrite the
congruence in (6.6.11) as
mxyh + n = 0(mod pz).
We take T=PQ62(2 + 6)p-1 and note that
Z (pzy1 I I 1
xyz — d n — Q/y m:P/x
{m,pz)= 1
for any €>0. Since
_, (ua\ \pz ^ Pz\a
\u\hpzii \Pz)~ 10 otherwise
Proof of Theorem 6.7
185
(with the convention that the value u = —pz/2 is omitted from the summation if pz
is even, so we sum over a complete set of residues exactly once; all summations over
u in the following will be as above but with the value u=0 removed), we then have
*i= Z (/»)-'! Z Z 4
xyz=d u n~Qly m:P/x \
u(mxyh + nf
pz
\ + 0{Qd'p-1). (6.6.12)
(m,pz)=l
We have
^..hS/* \pzj\ \Pzy J
\PZn~Q/y
and so the main term in (6.6.12) is no more than
X Xminf-^-Ji/r1)
xyz=du \PZy J
'u mxyh\
m:P/x
(m,pz)=l
pz
J
(6.6.13)
We may apply Lemma 6.6 directly to (6.6.13) for \u\^pzy/Q, and apply it in
conjunction with partial summation (or divide into blocks U^u<2U) for
\u\ >pzy/Q. The bound we obtain for (6.6.13) is thus
« I gl+«z-(l/2)J,-lp-(l/2)<<gl+«p-(l/2)d«
xyz = d
Hence, since d^Y forces (d,p) = l, we have
X \Rd\ti2(d)de«Y1 + 2'Q1+ep-<1/2)
(d,2h)=l
for any € >0. We may therefore take Y=P114, € = 1/100, and employ Lemma 6.5
to demonstrate that the number of solutions to
rp-sq = h, r:Q, q~Q, s:P
is
ce2Q( ih v _
*is&m)+olQ
19/2Ch
(6.6.14)
Now
so, summing (6.6.14) over h and p gives the bound (6.6.6), using the fact that
4 log P ^ log g. This completes the proof of Lemma 6.3. □
It should be noted that some averaging over one of p or q is necessary in the
above. The inequality (6.6.5) can have too many solutions for fixed p,q if plq is
near to a fraction with small denominator.
186 Diophantine approximation with restricted numerator and denominator
NOTES
In [112] the author first considered problems of this type for both variables
square-free, but did not get a 'best possible' result. Later, Vilchinskii [270]
obtained a qualitative best possible result for this situation when ij/(ri) =
0(w)(wlogw)-1, with 0(w) non-increasing. It would be of interest to prove a result
on the number of solutions to
|ma—n\<il/(m), m^N, m,n square-free
with a good error term (like the one obtained by the author in [118] for the case
of a prime and a square-free).
It should be possible to generalize the results of this chapter to higher
dimensions, for example to consider
«i-
< $i(q\
<*2~
Ei
Ei
q
with pl9 p2, q prime, when
y ^i(q)^2(q)
q prime *^o H
<<A2(tfX
diverges. Further generalizations might include linear forms in several prime
variables.
The reader will note that the results in Chapter 8 suggest other possibilities,
for example |r"a — p\ <(log«)_1 with p a prime and r^2.
It is possible to modify Lemma 2.3 and thus Lemma 6.1 to provide some
information on the number of solutions. To be precise, in many cases we can
obtain
where Jf(v.,N) denotes the number of solutions to (6.1.1) and
W)=2>(")iK").
nesJ
These cases include those discussed in Theorem 6.7. Similar comments will apply
to our subsequent applications of Lemma 6.1.
7
Non-integer sequences
The sequence anoc. Criteria for infinitely many solutions. Lacunary
sequences. The sequence (nQ + y)a + pn. Asymptotic formulae. Sequences
of- and a%. A generalization of GCD sums.
7.1 INTRODUCTION AND STATEMENT OF RESULTS
In this chapter we shall consider generalizing the results of Chapters 2, 3, and 4
to non-integer sequences. We may thus consider approximations to almost all a
by 'fractions' of the form mfan, where weZ but an comes from a given increasing
sequence of real numbers. It seems an abuse of terminology still to call this
Diophantine approximation (since Diophantus was concerned with solutions in
integers), but we hope that the reader will bear with us, since we know no better
term. We shall also consider when it is possible to obtain asymptotic formulae for
the number of solutions to inequalities such as
\\aaH + p\\<iKri), n^N (7.1.1)
or
How, + All <*(«), n^N, (7.1.2)
where p is a given fixed real and /?„ is a given sequence of reals. Under certain
circumstances our results in Chapter 5 for non-integer sequences will provide an
answer to (7.1.1), but often we shall be able to establish stronger results. This is
possible since the worst intervals for discrepancy may be quite long; the
distribution in very short intervals may be better.
From one point of view the fundamental question we address here is: what
properties of Z were vital in Chapters 2, 3, and 4? Factorization and coprimeness
played a significant part in our proofs, and we immediately lose these important
characteristics if an is an arbitrary real sequence. If we were dealing with a
multiplicative semigroup whose generators mimick the primes in their distribution, then
we would expect similar arguments to those we have deployed earlier to lead to
analogous results. In the general case our results could not be 'best possible', since
they would have to include the best results that can be established for integer
188
Non-integer sequences
sequences without the use of factorization and coprimeness. We shall discover (see
Theorems 7.1 and 7.2 below) that sometimes it is possible to retrieve most of the
advantages of an integer sequence by careful consideration of the remaining
integer variable m in mlan (that is, the implicit integer variable in the || • || notation).
As in previous chapters, the divergence of
X>(")
n=l
is a necessary condition, and so we henceforth assume this series diverges. We have
already demonstrated in Theorem 2.8 that this is not a sufficient condition.
There are certain sequences which spring immediately to mind as likely
candidates for an, such as/(«) where/is a polynomial with real coefficients (in Chapter 4
we dealt with integer, and so by a simple extension rational coefficients). However,
a moment's thought reveals that such an example may prove difficult even in the
linear case an=n6+y. If 6/y is very well approximable by rationals, then we are
dealing with a sequence which for much of the time resembles na+b with a,beZ,
and so the properties of integers will be required to obtain 'best possible' results.
On the other hand, if 6/y is not very well approximable, then the sequence should
be more 'random' in behaviour; see Theorem 7.4 below for this case. For
polynomials of degree at least two, Theorem 7.1 furnishes a nearly best possible answer.
For the case of lacunary sequences the problem was first discussed by Cassels [54]
and Philipp [214], while R.C. Baker [12] dealt with sequences for which
N
n=l
which are very nearly lacunary. We establish the lacunary case in Theorem 7.3
below. The general case was dealt with by the author in [121] and [123].
The problem in its most general form—approximating a by 'fractions' am/bn,
where am, bn are two given sequences of reals—is too difficult to handle. Clearly
there must be various cross-conditions linking am and bn (compare our discussion
in Chapter 6) to make the problem tractable. In the case when am, b„ take on all
the values of given polynomials at integer arguments, there is some hope (compare
the result at the end of Section 6.1). Also, if am, bn take on the values of
multiplicative semigroups, then more progress should be possible, provided that
the ratios ajan are 'sufficiently dense' in IR.
Theorem 7.1 Let an be a sequence of reals satisfying:
(i) an+! — an ^ X for some A>0, for all sufficiently large «; (7.1.3)
(ii) \an\>cnx +3 for some c, <5>0; (7.1.4)
(iii) am/an<Km~1n~fl for m^ri* for some k, \i, (7>0. (7.1.5)
Then, for almost all a, there are infinitely many solutions to
||a„a||<(«loglog9«) 1.
(7.1.6)
Introduction and statement of results 189
Remark Clearly (i), (ii), (iii) are satisfied if an =/(«), where/(w) is a polynomial of
degree at least two (replacing/by —/if its leading coefficient is negative), or indeed
any linear combination of powers of n, provided that at least one power exceeds
one. These conditions are also satisfied by many other sequences, for example,
an=f(pn), where pn denotes the wth prime and/is a polynomial of degree at least
two, or an=exp(np) for any /?>0 (although when p ^ 1, Theorem 7.3 gives a
complete solution to the problem). The significance of condition (iii) is to prevent very
large gaps between the an which are then followed by many relatively small gaps.
Condition (iii) is automatically satisfied when an«nA for some fixed A. The reader
should have no difficulty deducing the following corollary to Theorem 7.1 by taking
a subsequence of the an.
Corollary Let a„ be an increasing real sequence with
0<c^an+l-an<nA
for some values c, A. Then there are infinitely many solutions to
Rail < ii'"1
for almost all real a and every € > 0.
Theorem 7.1 is an immediate corollary of the following result. It seems no
easier to prove Theorem 7.1 directly at present.
Theorem 7.2 Given the hypotheses of Theorem 7.1, for almost all real a there are
infinitely many solutions to
|ana-m|<(«loglog9«)_1, (7.1.7)
where m has no prime factor smaller than n1 +€, with € = 1/4 min {a, S, jx). Moreover,
if |aj <nA for some fixed A, then there are infinitely many solutions to (7.1.7) with
mas a prime.
Remark We note that by Lemma 1.2 there are only finitely many solutions for
almost all a to
\a„a—m|<«"1(loglog9«)"1-''
for any 77 >0 with the above restriction on m, so Theorem 7.2 is very near to best
possible. Theorem 6.8 follows immediately from Theorem 7.2. In the following
results we write
y(A0=2>(«),
n=l
and we have already assumed that *P(oo) diverges.
Theorem 7.3 Let an be a lacunary sequence, /?„ a real sequence, and ij/(ri) a sequence
of positive reals. Then the number of solutions to (7.1.2) is
2¥(JV) + 0(¥(JVy/2(logW)3/2+€) (7.1.8)
for any € > 0.
190 Non-integer sequences
The next result generalizes Theorem 4.2.
Theorem 7.4 Let si be an infinite set of positive integers, and suppose there exist
constants au a2 such tnat
0<al^\l/(n)/\l/(m)^a2 when \<n<m<2n.
Write
V(J/,JV)=X*(i)
n=l
ties/
and suppose *P(^, oo) diverges. Let 6, y be real with 05*0, 6/y irrational. Then the
number of solutions to
||(«0 + y)a + jM <*(«), nest, n^N, (7.1.9)
is, for almost all a, equal to
2¥(j/, A0 + O(<D1/2(A03/2 +€), (7.1.10)
where €>0 is arbitrary . Here <b(N) = x¥{N) unless 6/y satisfies
liminf exp(c^)||^0/y|| =0 for every c>0. (7.1.11)
q-»oo
When (7.1.11) holds, one takes
N
G(iV)= I (log*)*(#)•
Remark We note that in the above, <b(N) = *¥(N) unless d/y belongs to a certain
set of transcendental numbers having Hausdorff dimension zero (see Chapter 10).
The problem with these numbers will become apparent in Section 7.4.
Theorem 7.5 Suppose that, for some constants c>0, <r<l, we have an+l—an>
cn~a. Let € >0 be given. Then the number of solutions to (7.1.2) is, for almost all a,
equal to
2V(N) + 0(Npi<T\\ogN)5l2+€). (7.1.12)
Here,
p((7) = i(l + max((7,0)).
Corollary lfan+1—an>cforsomec>0, and xi,(N)»N(1,2)+3 for some <5>0, then
the number of solutions to (7.1.2) is asymptotically equal to 2W(N)for almost all cc.
We now show that we can do better when the sequence a„ increases sufficiently
rapidly.
Introduction and statement of results 191
Theorem 7.6 Let € > 0, and suppose that an is a sequence with
log a
lim - = oo, and an+ x — an ^ 1 for all n. (7.1.13)
n-oo log"
Then, for almost all a, the number of solutions to (7.1.2) equals
2¥(A0 + O(<D1/2(A0), (7.1.14)
where
<*>(A0= 2>(")«€.
We note that Theorem 7.6 is almost as strong as Theorem 3.3, although we
have had to impose the condition (7.1.13). Our method of proof will follow
the same lines as the proof of Theorem 3.3 (Lemma 7.8 below substitutes for
Theorem 3.2).
Before considering a generalization of the problem and embarking on the
proofs, we mention that a common theme to the proofs for the above results is
the estimation of the number of solutions to inequalities of the form
<e,
which we have already encountered frequently when dealing with integer
sequences. As in previous chapters, the greatest difficulty arises when there is a
very good solution with r and s 'small', leading to many other solutions as
multiples of r and s.
Another way of generalizing the problems discussed in earlier chapters is to
consider
||/(«,a)||<iA(«), (7.1.15)
where / satisfies certain conditions on its derivatives with respect to a as n-*ao.
For example, Cassels [54] showed that there were infinitely many solutions to
(7.1.15) for almost all txe[a,b] when/'(«,a)>0 and satisfies:
(i) /'(«, a) -> oo with n for fixed ae [a, b];
(ii) /'(«, a) increases monotonically with a for fixed n;
(hi) ZmnTM-^oWHN));
n=l
and
(iv) X il/(n)En=o(V2(N)),
192 Non-integer sequences
where
E =
!*b n
n
a m= 1
A sequence of special interest would be a" for a>l. By the results of Chapter 5
we know that (7.1.15) has infinitely many solutions for almost all a>l when
^(w)=w~1/2(logw)5/2+€. Indeed, one could consider aa" or even a*, provided that
the sequences an were sufficiently well behaved. We expect to be able to do better
for small fractional parts, and the following results confirm this.
Theorem 7.7 Suppose that /?„ is a real sequence, and an is an increasing sequence
with aa+1—an»n~c for some c, and
a
n oo as «->oo.
log«
Then, for almost all col, and any €>0, the number of solutions to
||aa"+jW«K"X n^N (7.1.16)
is
2¥(A0 + O(¥1/2(A0(log¥W)3/2+e).
Remark This improves the results of Koksma [169] and Cassels [54], who
essentially needed an+1—an^c>0.
Theorem 7.8 Let /?„ be a real sequence, and an a positive increasing sequence of
reals with aJ_1«~3-> oo asn^co for some B>\, an+1—an>c for some c>0. Then,
for almost all real ae[2?,oo), the number of solutions to
(7.1.17)
is
2X¥(N) + 0(¥1/2(AO(log ¥(JV))3/2 +€),
for any € > 0.
Remark In both of the last two theorems, the nature of the problem changes
with the altered dependence on a. Essentially the problem loses its dependence on
arithmetic; instead it is the analytic properties that shape the sequences' behaviour.
The hypotheses on the sequence an can be relaxed in both these results.
7.2 PROOF OF THEOREMS 7.1 AND 7.2
In the following, we write [x] to denote the nearest integer to x (or, if {x} = J,
take {x} = [x]). Let 0<jS<y and put 0 = y-jS. We write
tj/(n)=min (1/12, l/(« loglog 9«)),
Proof of Theorems 7.1 and 7.2 193
E(m n)= ^m^a"
log 2m log In
and
<srf(m) = {ueZ: p\u=>p>m1+€}
(if \an\<nA, then we may take s£(m) to be the set of primes). We note that if
V>2m, then <tf(m)n[V,V(\ + 6)] contains BVQ{m,V){\ + o{\))\og~lm members,
where ^(1 + €)~x < Q(m, K)< 1, unless we take s/(m) to be the set of primes, in
which case Q(m,V) = (log m)/(log V). Also, we put
/ = (fty), L(A0 = logloglog(100A0
and
To prove Theorem 7.2 (and hence 7.1), we need to show that almost all a belong
to infinitely many <2)n. We shall use Lemma 6.1 for this purpose, so we must
estimate
AT
X^J and X K@nn<%m).
n=l l^m<n^N
As in Chapter 6, it is crucial that the main terms in our estimates have the correct
dependence on 6. For the sake of simplicity in the proof, we shall suppose that
the values A, c, and K in the theorem's statement are all equal to 1 and that y<\.
We have
n=l n=l an resf(n)
rean/
= 2y <K") an6Q(nJan)
= i an logn
in our above notation. Hence
N
9L(N)« X l(@n)« 9L(N). (7.2.1)
n=l
Here the implied constants are absolute unless one takes s£(m) to be the set of
primes, when they depend at most on A. This establishes (6.2.1) and supplies part
of the bound required for (6.2.2). To complete the proof, we need the following
lemma, whose demonstration we shall leave until after we have finished the proof
of Theorem 7.2. In the following, constants implied by the « notation depend at
most on a, S and \i. Constants implied by the o and O notations may also depend
194 Non-integer sequences
on /? and y. Hence the statement A«B(l + o(l)) does not contain a tautology:
it is vital, since we let N-+ oo and then 0->O in the proof.
Lemma 7.1 Suppose that m<n, and put A = A(m, n)=flm/fl„. HPirc'te iV(m, n) for the
number of solutions to
||rA||<2<Km), p\r=>p>nl+\ />|[rA] ^p>m1+\ reanf. (7.2.2)
(i) ifm^rf we have Af(m,/i)/£(/w,n) « 0(1 + o(l)); (7.2.3)
(ii) if «<T<m^«—«€ we have N(m,ri)/E(m,n) = 0(ioglog9n); (7.2.4)
(iii) if n—n€ <m<n we have iV(m,n) = 0(\l/(m)an/\og2ri). (7.2.5)
We complete the proof of Theorem 7.2 by noting that, for m<w,
/.(@nn@J^2—N(m,n).
By (7.2.3) we obtain
£ »(*) Ww,fi) g g " y, »(fi)»(iw)(l + o(l))
„=ii»- «« «-i«<«- log 2« log 2m
«9L2(N)(\ + o(\)).
By (7.2.4) we have
" y \lf(n)N(m,n) y ^(#i)^(/w)loglog9«
n=l nff^m^n — n*
fl» n% log 2« log 2m
nff ^ m ^ n — nc
" 1 1
„f!«log 2« n.^nm log 2m loglog 9m
« I i o I i Q <<ZW
m «! m log 2m loglog 9m
Also, by (7.2.5) we obtain
» Hn)N{m,n) g »(n)»(m) ^
n=l 0<n-m<n* fln nsgJV lOgZW
0<n-m<n«
We thus obtain
£ MHn0m)« dL2(N)(\ + o(l)),
1 <m<n^iV
which with (7.2.1) establishes (6.2.2) and so completes the proof. □
Proof of Theorems 7.1 and 7.2 195
Before proving Lemma 7.1, we need three further lemmas. We write
and note that W(m,v) ^ v/<p(v). Henceforth we fix s/ to be {weZ: p\u=>p>m1 +€}.
The reader should have no difficulty in making the necessary alterations when jtf
is the set of primes.
Lemma 7.2 Let a, q, m be positive integers, and b a non-zero integer, with (a, q) = l,
q>2,b^0 (mod q), and (b, aq)esf(m). Let B, C be positive reals such that C > nfq.
Then the number of solutions to
ra = b(modq), r(ra-b)/qejrf(m), B^r^B + C (7.2.6)
is
CWjmMq) n 0 7,
q logz2w
Proof The solutions to (7.2.6) have the form
r=s(b,q), (ra—b)/q = (a,b)t, stes#(m),
where
ab hq be ha
s=-—r + -T^-, t=- - + •
(b9q) (b,qY (qa,b) (a9b)9
aa-\ B-ab , B + C-ab
q/(q,b) q q
Here a denotes the inverse of a (mod q/(q,b)). The bound (7.2.7) the follows from
the two-dimensional sieve. For example, it can be derived from Theorem 5.1 in
[108] in a similar manner to Theorem 5.7 there. The important point here is that
h ranges over >m€ consecutive integers, which enables one to obtain the correct
formula for the number of solutions to st = 0 (mod d) with a smaller error term
for d<m*12, say. This is the information required to estimate the remainder term
in the sieve (compare Section 6.6, where we did this for the three-dimensional
sieve). □
Lemma 7.3 The number of integers r in an interval [B,B + C) with restf(m) is
« C(log 2m) ~ * provided that C ^ m€.
Proof This is simple upper-bound sieve result (compare Lemma 6.4) and follows
from Theorem 2.2 in [108]. □
196 Non-integer sequences
Lemma 7.4 Let k be a positive integer, and C a real with C^l. Then
I ^ = C^n(l + ^\rVtf(^)log2C). (7.2.8)
Proof This elementary result is a variant of Lemma 2.5, and may be obtained
with a suitable alteration to the proof there. □
Corollary // C>ky for any fixed y>0, then the left-hand side of (7.2.8) is
«C(p(k)lk.
The corollary follows immediately upon using the standard bound for the
divisor function.
Proof of Lemma 7.1 We first consider case (I), n^m*. We put A = 2ij/(m) and
write y = A "*. Then
|| rA || < A => re(sy—Ay, sy+Ay),
and in this lemma we have restf(ri), ses/(m). Thus, by Lemma 7.3 (which is
applicable is view of (7.1.5)), for each s there are «,4j>/log2w values r. Also, the
variable s lies in the interval amf and so, by Lemma, 7.3, takes on «am9(\ + 0(1))/
log 2m values. Hence, in this case,
N{m,n)« ^*l + *l)) =2Bi\ + o{\))E{m,ri),
\og2m\ogln
as required.
To consider cases (II) and (III) we note that, by Theorem 2.1, we can find a, q
with
A=- + p, (a,q) = l, q<ann-\ qan\p\^n*. (7.2.9)
We now split the argument into four cases depending on the sizes of q and /?.
(i) The case q = \. We note that &~^nalan and so a #0. Thus a = \ (as am<an),
which forces an—am<n* from (7.2.9). In view of (7.1.3), this indicates we are
dealing with case (III). We wish to bound the number of solutions to
||r/y || <A, where y=anl(an-am).
Here r lies in intervals of the form (ty—Ay,ty + Ay), with / = 0,..., \an—amJ. We
have yA^a„n~€/(n\og\og9n)»n€. Thus we can apply Lemma 7.3 to obtain the
bound
^(fl«-°Io^ = loi2P
as required.
Before considering the other cases, we first investigate the possibility that ra=0
(modq). Since restf(ri), this eventuality arises only when q^n1+€. The number of
Proof of Theorems 7.1 and 7.2
197
solutions is then trivially «an/q = o(E(m,n)). We may therefore assume that ra^O
(modq) in the following discussion.
(ii) The case \<q<(2A)~1. Without loss of generality we may suppose that
p>0. If2qpa„<\, then the only solutions will have ra = 0 (modq), which we have
already considered. If 2#/fa„^l, then
rA||<^
ra t
— + -
q q
<2A=>ra=—t (modq).
Here t takes positive integer values not exceeding 2qa„p, and, for each value of /,
the variable r is restricted by
(t/q -A)P~1^r^ (t/q + A)p
-l
Since Aft l ^ qaji €/(m log log 9m)» qm€, we may apply Lemma 7.2 to obtain the
bound
N(m, n)«
aq
tAfi
-l
v&q) t^an <pV)q iog22w
«E(m, n) loglog 9« (7.2.10)
by Lemma 2.5 (that is, Lemma 7.4 with k = \). We have also noted that aq«n3
in this case, so that aq<p~ 1(aq) «log\og9n by the standard bound for Euler's
function (Theorem 328 in [111]). We remark that the factor aqq>~x(aq) cannot be
cancelled out in (7.2.10) by adding the summation restriction (t,aq)estf(m), since
the summation range may not be long enough to apply Lemma 7.4 with
k = K(aq,m) = Y\ P-
p\aq
p<m1+*
(iii) The case q'^(2A)~1, p'^-A^a^ We remark that these conditions imply
that q«n1 + 2€. We split the interval [l,a„] into subranges of the form
[hp~1A,(h + \)p~1A], with h = 0,\,...,lpanA]. To consider the solutions for r in
one such range, put t = t(h) = [(2h + \)qA/2]. Then
|rAII<i4
ra t
— + -
a q
<3A.
Hence it suffices to bound the number of solutions to
ra = b-t(modq), \b\^3Aq, b^t, r(ra + t-b)lqe$t(m).
Since p lA»qm*, this number is, by Lemma 7.2,
«
p-lAW{m,aq)
\b-t\
q\og22m W£AQ <p(\b-t\)
i>#r, (b-t)esf(m)
(7.2.11)
« _i. 2^- —7—; L -TIT «E(m>n)logloS9n
198 Non-integer sequences
Summing (7.2.11) over h then gives a bound
jg"1^ ag y _b_
q \og22m <p(aq) b£aJiq <p(b)
by Lemma 2.5 (or 7.4 with k = 1); we note again that aq « n3 in this case and thus
obtain the required bound for N(m,ri).
(iv) The case q^ilA)'1, fi~1A>an. In this case \rp\<A, and so
\\rA\\<A=>\\ra/q\\<2A.
We therefore need only consider solutions to
ra=b(modq), \b\<2Aq, 6#0, r{ra—b)lqestf{m).
An application of Lemma 7.2 then yields
\u \ anW(m,aq) b ,n <* m
N(m,n)« — X -7K- (7.2.12)
?log22m b£M <p(b)
(na,q)esif(m)
If q^n3/2, we apply Lemma 7.4 with k=K(aq,m). Since k^aq<q2 and Aq^
q/(nloglog9ri)»qn~5l*>q116, we obtain the bound N(m,ri)«E(m,ri), and this
completes the proof. We note that we did require the extra saving from the
summation restriction here, because aq(p~x{aq) could be larger than log log 9« in
magnitude if the sequence an increases rapidly. □
7.3 A REDUCTION OF THE PROBLEM AND PROOFS FOR
THEOREMS 7.3 AND 7.5
We work with the measure
j"OT =
7c1/2exp(-7c2jc2)d>l
xeS
defined in (5.3.1). We note the following result which replaces the standard appeal
to Parseval's identity in periodic problems.
Lemma 7.5 Let f, g be functions with period one, both square-integrable on [0,1),
having Fourier coefficients /(«), g{n) respectively. Then, if a and b are non-zero
real numbers, we have
f(ax)g(bx)dn= £ f(m)g(n)exV(-(am + bri)\ (7.3.1)
'R m,n= — 00
where the double sum on the right of (7.3.1) is absolutely convergent.
We note that in Theorems 7.3-7.6 we have an»n€, and we may assume that
ax >1. We let/„(a) be the characteristic function of those a with
ll«fl.+AII<*(i),
A reduction of the problem and proofs for Theorems 7.3 and 7.5 199
and put/n = 2^(«). We shall want to estimate
n=l
via Lemma 1.5. In the following it will be tacitly assumed that the appropriate
integer variables are restricted to s# when proving Theorem 7.4, and all integrals
are over coeR.
We have
Z »)- Z fn\&ii = I2-2SIx + S\ (7.3.2)
^t<n^u t<n^u
where
h-
£ /.(<») )dfi (7 = 1,2),
Kt<n^u
and
5=2 X iK«).
Now we have, in the usual L2[0,1) sense for Fourier series,
fn((o) = 2il/(n)+ £ cke(kanco\
k*0
with ck«min(il/(ri),k ). Thus Lemma 7.5, with g(x) = l, gives
oo
Let
Ii=S+0( Z ZminOK»U-1)exp(-A:20n2)
^t<n^uk= 1
M= max \j/{n).
t<n^u
Then, since
00
Zexp(-fc202)«exp(-a2)«w 2,
jt=i
we have
/1=,S+0(M). (7.3.3)
To estimate I2 we write/„+(a>) for the function constructed in Lemma 5.1 with
J = (-y-il/(n), -y + \Kn)) (interpreted (modi)), L=[«3/2]. Write g(n) = 2il/(n) +
(L + l)_1and
hn(cD)=f;(co)-g(n).
200
It follows that
Non-integer sequences
h^
I /?(») in
^t<n^u
= £ gin)) + 2 X g(«)
,r<n^M
r<n^M
Z hm(<»)&H
t<m^u
+
Z ^m(w)) <ty-
vl<m<u
The first term in (7.3.4) equals
s2+o(s+ x «"3/2)-
\ t<n^u /
Working as we did to obtain (7.3.3), the second term in (7.3.4) is
«M(S+\).
Finally, the last term in (7.3.4) is, by Lemmas 7.5 and 5.1,
Z (p(m,n),
Km^n^u
where
(7.3.4)
(7.3.5)
<p(m,ri)= Z minGK#i),|r| ^min^w),^! 1)exp(-(A:am + ran)2). (7.3.6)
k,r*0
Combining (7.3.2)-(7.3.6) then gives
f I fjfo)- I fXdf
« z <K«)+ Z <?("*>")+ Z «~3/2-
t<n^u t<m^n^u
t<n^u
In order to apply Lemma 1.5 we therefore need to obtain bounds of the form
Z (p(m,n)« Z <Pn
t<m^n^u t<n^u
for some q>n. We do this by reducing the estimation of (7.3.6) to the problem of
A reduction of the problem and proofs for Theorems 7.3 and 7.5 201
bounding the number of solutions to certain inequalities. We can do this since
X <p{m,ri)K f>p(-(/z-l)2)
* Z Z Z mmmnWrr^mmWmAkr1). (7.3.7)
\k\^mil2\r\^nil2
\kam-ran\<h
We have thus reduced the problem to estimating the number of solutions to
the familiar inequality
\kam-ran\<h.
We note that using a trivial bound, we may assume that h ^ log u.
Proof of Theorem 7.3 We have a„+1> can for some c> 1. We then consider
For all large n we have h/an<n~3, and so at most one fraction r0/k0 with
(>*o>^o) = l wiH satisfy (7.3.8) for each pair (m,«). Every other solution will then
be of the form (r,k)=s(r0,k0). Also we note that since r0#0, we have k0^
\r0ajam. Thus the number of solutions to (7.3.7) is
oo
« Z Z min(^(H),|a,,r0j/allir1)min(^(/fi),|rasr1).
Km^n^u s= 1
We estimate the above sum by treating the cases s ^ \li(n)~x, s>if/in)'1 separately.
This leads to the bound
« Z (*(*) + *("))—« £ <Kn),
a
as required to apply Lemma 1.5 and obtain (7.1.8). We remark that, as noted by
R.C. Baker [12], all that we needed essentially was
N
5>„«%. □
n=l
Proof of Theorem 7.5 We consider the number of solutions to (7.3.8) now by
fixing n,r, and k and counting solutions in m. Also we will take the M-1,!/:!-1
terms in the minima in (7.3.7). Clearly there will be no solutions if k^-r+h,
because the an are increasing and m^n. On the other hand, if k <r+/z, there are
no more than
2hn° ,
+ 1
cr
h
<—.
a„
(7.3.8)
202
Non-integer sequences
solutions of (7.3.8), since am+1—am^ cm a^-cn a. Hence
00 1 (hna \
X ?(m,*)<8£exp(-(A-l)2) £ tt — + 1 « I "^"'log2*.
Km^n^u h=l t<n^u^r \Cr J t<n^u
k^r + h
An application of Lemma 1.5 then yields (7.1.12) as desired. □
7.4 PROOF OF THEOREM 7.4
Clearly we may assume that 0 = 1. We shall also suppose that (7.1.11) is false for
all sufficiently large c. It should be evident from our proof that when (7.1.11) holds,
the results will be worse by only a logw factor. At the end of this section we shall
discuss the problems caused when (7.1.11) holds. It should be noted that we have
not previously proved a formula for the number of solutions to (7.1.9) in the case
y = 0 unless /?„ is constant (see Chapter 4; Theorem 3.1, which does have /?„,
requires a faster growth of *F(iV)). However, it seems to be no easier to consider
/?„ fixed in the present context. The bound for the number of solutions to the
fundamental inequality is contained in the following result, whose proof we defer
until after we have completed the proof of Theorem 7.4. In this chapter the
notation a ~A reverts to its usual meaning in this monograph, that is, A ^ a<2A.
Lemma 7.6 Let h,K,L,M,N>l, and suppose that y is a real irrational with
\\qy\\>A-<,*XLq>l, (7.4.1)
for some A. Then the number of solutions to
\l{n + y)-k(m + y)\<h (7.4.2)
with
k~K, l~L, m~M, n~N
is
« h2 min(K, L) max(M, N){\ + |logGR7L)|) + (K+L)3'2 +€ (7.4.3)
for any e>0. The implied constant depends only on A and e.
Remark We note that the result is false for rational y, as there will be too many
solutions to
l(n + y)-k(m+y) = 0,
and so (7.4.3) will fail for those y with extremely good rational approximations.
We have already seen the problems caused by this phenomenon in previous
chapters.
Proof of Theorem 7.4 203
Proof of Theorem 7.4 Write
n
<Pn= Z <P(m>n)'
Then
Z q>(m,n)^ Z <P*
The reader will note that, although we appear to have thrown away a great deal
in terms of the short interval estimate by allowing m to run over all values less
than «, Lemma 1.5 requires only the long interval estimate, which has not been
compromised by our approach. We have
Z (p*«Zexp(-(/*-l)2)
n=l h=l
x £ r(ilf,JV,^,L,A)min(^(JIO,^"1)min(^(A0,L"1),
M,AT,K,L
where M,N,K,L take values of the form 2j with M ^2N,N ^2X,K^2M312,
L ^ 2Af3/2, and Y{M,N,K,L,h) represents the number of solutions to (7.4.2). For
given M,N,K we observe that L can take on only 0(\og2h) values for which
Y(M,N,K,L,h) is non-zero. Write
Y\M,N,K,h)= max Y(M9N9K,L,h).
L^(KM + h)/N
Then
Z^«Zexp(-(A-D2)
n=l /i=l
x £ /zr(M,A^,^,/z)min((A(M),/i:-1)min((A(A^),Ar(max(l,^M-/j))-1)
M,N,X
« Z MKil + llogiMlNWmmiil/iMXK-^mmiil/iNXNiKMy1)
M,N,K
by Lemma 7.6. Here we have noted that values of h>\KM make a negligble
contribution to the sum, and (L + K)5I3«KM (that is, we have taken e = l/6 in
Lemma 7.6). Carrying out the summation over K first in (7.4.5) yields a bound
« Z M(\ + \og2(M/N))il/(N)« ZMKAO « Z MO-
M,AT N n=l
This completes the proof, in view of Lemma 1.5. □
Before we prove Lemma 7.6, we require the following result.
204 Non-integer sequences
Lemma 7.7 Let l,T, h> 1, and suppose that y satisfies (7.4.1). Then the number of
pairs of integers b, t with
t~T, l\t, l\([ty] + b), \b\^h
is
«h(Tr2 + T(l\og2T)-1), (7.4.6)
where the implied constant depends only on A.
Remark If y does not satisfy (7.4.1), then one obtains hTl'1 in place of (7.4.6).
Proof By Theorem 2.1, and making use of (7.4.1), we deduce that there are
coprime integers a, q with
log 2T« q ^ 77(log IT) (7.4.7)
and
|y-a/^l<aog2D%r)«r-1.
Here ty = taq~1+0(l). We then wish to count solutions to
ual + r=clq
with
T^ul<2T, \r\^(h + 0(l))q.
This leads to the congruence
ua = r(modq), \r\^(h + O0))q/l.
The value r = 0 gives a contribution ^:2T(lq)~1, since u must then be a multiple
of q. The other values of r give
-*(■♦£
solutions, and so (7.4.6) follows from (7.4.7). □
Proof of Lemma 7.6 We suppose, without loss of generality, that K^L. We also
assume that A/>M; the proof works equally well for M<N. Put
t = k—l, y=n—m.
We are then concerned with bounding the number of solutions to
\ly-tm-ty\<h. (7.4.8)
We note that the contribution to the number of solutions to (7.4.2) from cases
with either one or both of y, t zero is « hLN. We split the remaining values of y
and / into blocks as usual of the form y~Y, t~T. Put g = (t,y). We then treat the
cases 2g^TL~1i2g<TL~1 separately.
Proof of Theorem 7.4 205
(i) The case 2g^TL~1. We have g\([ty]+a), for some a with \a\ ^h + \, and
count solutions to
lyg = {[ty]+a)g-xmod{tg-l\ \a\Kh + \. (7.4.9)
By Lemma 7.7 there are «h(Tg~2 + T(g\og2T)~1) pairs t, a for which this can
happen. Of course, there are «Yg~1 values of y available, and once a, t, y are
fixed, then (7.4.9) has ^ LgT~x + 1 < 4LgT~1 solutions in /. Since g ^ 2 min (T, T),
the number of solutions with 2g^TL~1 is
,<2«±(y,T) VT/WV^2 0 log 27
(ii) The case 2g<TL~1. Now we put/= (/,tg~l) and observe that
/^l([?y]+fl) for some a with |a|<A + l.
By Lemma 7.7 there are
/ T T \
<<{(M'2+f^2T)
pairs a, t for which this can happen. We are then left with the consideration of
wi; = jc(modz) (7.4.10)
subject to
m/~L, vg~Y, x,z fixed, (uv,z) = 1.
We also have (jc, z) = 1 and/#z ~ T. Now if Yf» T, then it is elementary (just count
the number of solutions in v for each u) that (7.4.10) has «YLT~1 solutions, which
we shall later demonstrate is an adequate bound. We henceforth suppose that
2fY<T, and note that the number of solutions to (7.4.10) is no more than the
number of solutions to
vx
<2LgT~\ (7.4.11)
Here v denotes inverse (modz). The number of solutions to (7.4.11) may be
estimated by Lemma 5.1 as
LYLg
ly /r\1/2+€/2
£ eIt
(v,z)=l
with an application of Lemma 6.6 to estimate the exponential sum. The number
of solutions in this case is therefore
206 Non-integer sequences
<K ,<TL-/<k- \T + \fg) hfg)1 +fg\og2T)
« h(LY log {TIL) + r<3 +€)/2).
Combining our estimates, the number of solutions to (7.4.1) is
«hLN+ £/*(Ly(l + max(0,log(r/L))) + r(3+€)/2). (7.4.12)
Y,T
Now, for each value of Y, T can take on only «(1 + log/z) ^ h values and (7.4.9)
still have a solution. Also, 7 takes on «log2iV values 1,2,4,... less than IN. We
conclude that (7.4.3) follows from (7.4.12), since
r(3+€)/2iog2Af«Ar+r3/2+€. □
Finally we compare the above process with the work in Chapter 4 (where /?„
had to be fixed). The irrationality of y has led here to a different mechanism for
controlling the size of the overlap estimates. It is not clear how one might try to
combine the two approaches for those y with very good approximations. Certainly
Schmidt's method can be adapted to deal with different rational approximations
to y, but these must be even better than suggested by (7.1.11) for the method to
work. It might be expected that there is a way to plug the gap, at least when the
sequence /?„ is constant.
7.5 PROOF OF THEOREM 7.6
The crucial part of the proof is the following lemma and its corollaries, which
generalize work we did in Chapter 3.
Lemma 7.8 Let real numbers F, G, R, S, U, V, be given, each no less than one.
Let rj>0,h^lbe given, and suppose that (7.1.13) holds. Then the number of solutions
to the inequality
\arv-uas\<h, R^r<R + F, S^s<S+G, u~U, v~V (7.5.1)
is
«Zh2(FGUV){1/2)+<. (7.5.2)
Here,
Z = max{«: an<n8T,~2},
and the implied constant in (7.5.2) depends on rj only.
If we replaced ar with any subsequence, we would not increase the value of Z.
Hence we obtain the following result.
Corollary 1 Let stf and $8 be any two finite subsets of {afj ^ 1}, n > 0, U, V, h ^ 1.
Then the number of solutions to
\arv—uas\<h, arestf, ase&, u~U, v~V
Proof of Theorem 7.6 207
is
«Z(\sS\\@\UV)ill2)+'h2,
where the implied constant depends on rj only.
From Corollary 1 we may then obtain the following result just as we deduced
Theorem 3.2 from Theorem 3.9.
Corollary 2 Let /(«) be any non-negative function, and suppose that R, S, h, t, u
are given, each being no less than one, and with t<u. Then we have, for every n>0,
Z Z f{n)f(m)«Z(RSri2)+*h2 £ f\n)n\
Km^n^u s*S t<m^n^u
r~R
\ams-aj<h
~n** mi
Proof of Theorem 7.6 Since the sequence an is 'given' and satisfies (7.1.13), we
shall henceforth absorb the 'Z' of Lemma 7.8 into the implied constants. Upon
applying the inequality min(A,B) ^ (AB)1'2 to (7.3.6), we obtain
I rf^K^aqK-CA-l)*) I (mm)m. (7.5.3)
\a»r-saj<h
To bound the sum over m,n,r,s on the right-hand side of (7.5.3), we first divide
up the ranges over r and s into subranges of the form r~R, s~S, where R and S
take on the values of powers of 2 not exceeding 2u3/2. To obtain an upper estimate
for the sum, we may include a factor
„3m3\»,
2o2 / '
R2S
where n = €/H, and dispense with the conditions r^«3/2, s^m312. Thus the
right-hand side of (7.5.3) is bounded by
Zexp(-(/z-l)2)Z(^5)"2""(1/2)Zw3''"3,7('A(w)(A(«))1/2, (7.5.4)
h R,S m,n
r,s
where we have omitted the summation ranges for clarity. We can now apply
Corollary 2 of Lemma 7.8 to the innermost sum in (7.5.4) with/(/i) = «3,ty1/2(/i).
Thus our estimate becomes
oo
« Z^2exp(-(/z-l)2)Z(^S)-'7 Z Hn)nlr>« Z <K")"€/2- (7.5.5)
208 Non-integer sequences
This enables us to apply Lemma 1.5 to obtain (7.1.14). The reader will have no
difficulty in verifying that
N \l/2 / / N \\2 / N \l/2
£ ft"* iA(«) j f logf I «(1/2)€ *(n) J J «f I iA(")"€
as required to complete the proof. □
Proof of Lemma 7.8 Suppose that Lemma 7.8 is true when U=V, R=S, F=G.
We now deduce the general case. Clearly the number of solutions to (7.5.1) is
oo
t= — oo\ v~V J\ u~~U
\th-arv\<2h \th-asu\<2h
R^r<R+F S^s<S+G
<fzf i 017 if s ^m
\r=-oo\ v~V / / \r=-oo\ u~U
\th-arv\<2h \th-asu\<2h
R^r<R + F S^s<S+G
<s
|flri; — asw | < 4/i |flrM — asw \ < A-h
R^rys<R + F S^r,s<S + G
«Zh2(UVFG){ll2)+\
by our assumption. We may therefore suppose henceforth that U=V, R=S, and
F=G. Clearly we may also assume that F^R, otherwise we can use a splitting-up
argument together with the result for F^R.
Let a = 2rj~1 + 4. Since the result is trivial for /y ^ J, we can assume rj<j in the
following. At times we shall write 9 or £ for real numbers with modulus less than
one, not necessarily the same at each occurrence. Thus 'A = B + t,C is equivalent to
*\A—B\ <C, for example. If R < Z, then the method used in proving Theorem 7.5
gives a bound «hF2V ^ h2ZFV. We can now therefore assume that R>Z, and so
ar>r8"'2^r(r+1)ff,
where T=[rj]~1 + \. We may also suppose that F>27<7, for otherwise there will
be only «Rh solutions.
If R<Vn, or h>R, we employ a trivial bound to establish the result. Fixing
r, 5, and u, then counting solutions in v, gives a bound ^ F2(2h + 1)K< 2>h2(FV)x +ri,
which is the required estimate. Henceforth we can therefore assume that R > V,
h<R, and soa)i~1>V°.
Now if (7.5.1) holds, we must have 1/3 ^ ajas ^ 3 (in view of our suppositions).
Write
*?k = {ar:3k^ar<3k+2,R^r<R + F}.
Proof of Theorem 7.6 209
Then, since
oo oo
fc=0 fc=0
we can assume that F is so small that arestfk for one fixed k, when R^r<R + F.
Relabel this set as jrf0 and put 3>0 = 0. We are now going to produce an analogue
of the argument used in Chapter 3. We divide s/0 into subsets Q)^ ^i such that
if r, s give a solution to (7.5.1), then ar, ase@p each ^ is 'not much bigger' than
the corresponding Q)}\ stf is the disjoint union of the Q}^\ and we can establish a
technique for bounding the number of solutions to (7.5.1) when ar, as are restricted
to @j. We shall be repeating earlier arguments here, but the details become rather
more cumbersome, since we are dealing initially with inequalities rather than
equations.
We define s/p @p Q)i inductively as follows. Let jrfj=<srfj_l\@j_l. If s/j is
empty, the process stops. Let ay be the smallest member of stfj, and write
s/j(t) = {arestf. \ayn-mar\<2T~anh, 1 ^m,n^(2V)\(m,n) = 1}.
Clearly, for some non-negative integer K^we must have
Now put
and let the process continue. Of course, this argument terminates after at most F
steps.
Now suppose that
\ard-asc\<h, (7.5.7)
where c,d~V. By the above division of js/0, there exist integers j,k with
are@pase@k. Without loss of generality, suppose that k^j. We shall demonstrate
that ase&j. Let ay be the smallest member of ^-, and t the integer picked satisfying
(7.5.6). Then
a1_a1aL_fm 0h2T-a\fc £h_
as aras \n ar )\d Vas
for some integers m, n with 1 ^ m <« < (2Vy. Hence
ay mc
as nd
KIT-'fca. mil*-1 h
"S
+ 7^" +
as \dar nV Var/
210
Non-integer sequences
Now ajar < 9,eld ^ 2,mln ^ 1,ar//z>Fff, V>2T, and so
/z27r
Ac
where x = t + \ — a, k=mc(mc,nd)~1, l=nd(mc,nd)~l. As kJ^(2V)t+1 and
(A:, /) = 1, this means that ase@j. We therefore deduce that the number of solutions
to (7.5.1) is no more than
22>(y),
(7.5.8)
where N(j) counts the number of solutions to (7.5.1) with are@j,ase@j.
Now suppose that ase@p areQ)^ so that
a
a.
m
n
2T~lh
a.
a.
u
as v
2Th
a.
with 1 ^ m<n < (2V)\ 1 ^ u < v ^ (2K),+\ (w,w) = (w, i?) = 1. We then have
Or = tal\~1(al\(m + e27
^/zVYw QTK
a„
a.
,ar
n
a.
v
a„
_ n_ merr^h\(u £2izhy
jn
a.
v
ac
since
and
Hence
n < lOw, 100 x 27*+* ha~ 1 < 1/3
i\+a)-1 = \ + ^6al2 for |a|< 1/3.
ar nu 0/z27t+3
as mv
ar
= ^ + 0(2K)
mv
-2t-3
Thus, if (7.5.7) holds for some c, d~V, then
nu c
mv d
da,
+ {2V)
-2t-3
<(2V)
-2t-2
(7.5.9)
As mvd^(2V)2t+2, we must have nud = mvc. We may therefore bound N(j) by
associating a pair of coprime integers m, n with each arG2)j or ^ and counting
solutions of nud=mvc with c,d~F, {m,n)GSx, (u,v)e&2. Here <fx consists of
points in ^*, the set of primitive points in Z2, associated with @p &j is the
corresponding set for #.. Let (p: ^-xf 2 be the mapping given by
ar(p = (m,ri)(e0>), for are@jy
Proof of Theorems 7.7 and 7.8 211
where m, n are the integers with
\ay-mar\<2Tnh, l^m,n^(2V)t+1, (m,w) = l. (7.5.10)
We then note the following properties of the mapping <p\
1. q> is well defined. To see this, note that if mv^nu, and 1 ^ u, v ^ (2V)t+1, then
2/z27t+1
m u
n v
^(2V)-2t~2>
a
Thus (7.5.10) has only one solution in m,n.
2. There are no more than h solutions in r to ar(p = (m,ri), since the argument
leading to (7.5.9) shows that ar(p=asq> implies
as as
and ar+1 ^ar + \ by (7.1.13).
The same results are true a fortiori for the mapping from ^ to Sv We
conclude that N(j) is no more than h2 times the number of solutions to nud=mvc
with c,d~V,(m,n)e$1,(u,v)e<f>2-
The proof of Lemma 7.8 is concluded by an application of Lemma 3.5, which
gives
^(7)«/z2(|^||^|)1/2Kexpf1 3/^^rg+f
loglog((2F)r+1 + l),
«h2(\9j\\^j\)ll2V1+\
Now
1/2/ \l/2
by the construction of #.,i^.. Hence the expression (7.5.8) is «(FV)x*nh2, as
required.
7.6 PROOF OF THEOREMS 7.7 AND 7.8
We can no longer use the measure \i. Instead, we revert to a more conventional
approach of initially restricting our interest to an interval [A, A +1], where A ^ 1.
We first require a variant of Lemma 5.3.
Lemma 7.9 Suppose that f(x) is a real-valued function with a monotonic continuous
second derivative for xe[a,a + l]. Then
P'+,^))dx<c ^ (min^j^,!)} (7.6.1)
212 Non-integer sequences
Proof Write b=a + \. If f\x)2>\f'\x)\ throughout the interval [a,b], then
this is just Lemma 5.3. If the right-hand side of (7.6.1) is 1, then there is
nothing to prove. Otherwise, suppose that/'(«)<(), f"(a)>0, f" is non-decreasing,
f'(x)2 >/"(*) for a ^ x<c, f'{x)2 <f"{x) for c <x < b (the other cases will follow
similarly). We shall show that the left-hand side of (7.6.1) is «/"(c)"(1/2) = €, say
(note €<1), which will complete the proof. By Lemma 5.3 the integral over [a,c]
is no more than e2. Also, by the mean-value theorem and the monotonicity of
/", we have
fix) ^f'{c) + {x-c)f\c) for jc>c,
^ -€_1 + (jc-c)e"2.
Hence/'>€_1 for x>c + 2€. Thus, using the trivial bound for the integral over
[c,c + 2€], and Lemma 5.3 for [c + 2e,b] (with obvious changes if b<c + 2e), we
obtain the required bound. □
Proof of Theorem 7.7 Working as in Section 7.3, the crux of the problem is to
estimate
PA+l
X hm(co)\ dco.
We note that, although we took T=[n312] in the definition of ff((D), we could
have taken T=[n1+Tl] for any n>0, and this we now do. In the following, the
summations over r and s are thus for \r\ <«1 +n, \s\ <mx +n. To prove Theorem 7.7
we must consider
X XminGKwMrr^minGKmXIsr1)
/M+l
e(ra°"—s<x"m)d<x
(7.6.2)
First we note that if an+l— a„>£n~c^ for w^l, and m<n, then either
\ranoca"~1 —sam(xa-~l\ >\£ma~~ln~c
or
\ran(an-\)<xa--2-sam(am-\)oca-'-2\>i€ma--1n-c.
To see this, note that when the first inequality fails, we have
samaa"-J =raX"~1 0 + 6en~ca-x),
where |0| <\. Thus
|ran(an-l)aa»-2-5flm(flm-l)aa«-2| = |raX""2((^-l)-(^-l)(l + ^«-c))l
Notes 213
and
{an-\)-{am-\)(\ + ezn-c) = (an-am)-(am-\)a;x6zn
>en c — ^€n c.
It follows that the expression in (7.6.2) is
« X \J/(n)n2+TI+c\ognA-{ll2)a- (7.6.3)
after an application of (7.6.1). Since the expression in (7.6.3) is bounded
independently of t and w, we may appeal to Lemma 1.5 to complete the proof. □
Proof of Theorem 7.8 We suppose here that A^B + e, and r]<d4. The proof
follows a similar pattern to that above, except now we note that, for an ^ e, at
least one of the following inequalities holds:
| raan log an -sa*m log am \ > \ craan ~ \
\raan\og>2an-saam\og2am\>icra*-1.
To see this, assume the first inequality fails and note that \ogan^\ogam + jca'1.
Then
I raan log2an -saam \og2am \ = \ra* log an log am -saam \og2am
+ raan\ogan(\ogan-\ogam)\
^ jcractn~l\ogan—^cratln~x'^\crcfn~x,
as desired.
The sum of integrals we must then estimate is
n^+^'2\ogn
« X <K") -r ,
with
C = i(i4 + €-l).
This expression is also bounded independently of t and u, and the proof may be
completed with an application of Lemma 1.5.
NOTES
R.C. Baker [12] actually dealt with the more general case of differentiable
functions gn(x) on a finite interval with
N
n=l
214
Non-integer sequences
He also showed that one could not replace (7.1.2) with
{anoc}eJn,
where Jn is a finite union of intervals for each n with A(^n) = \J/(ri) (of course, it
follows from our results that one could do this for Jn as a bounded number of
intervals for each «).
It would be of great interest to replace (7.1.13) with a much milder condition
and so improve Theorem 7.6. Theorems 7.7 and 7.8 have not been published
previously. There is still some scope for improving these results, and generalizing
all the results of this chapter to higher dimensions.
8
The integer parts of sequences
Prime values of [na] and [pa]. The sequences [otan], [a0"], [djj].
Asymptotic formulae in the lacunary cases and prime-normal numbers.
8.1 INTRODUCTION AND STATEMENT OF RESULTS
In the previous two chapters, we have begun to consider the distribution of [an<x],
where an is a given sequence. In Chapter 6 it was a deliberate decision to restrict
both numerator and denominator in our approximations. In Chapter 7 this was
merely a technical device to avoid many fractions coinciding. Nevertheless, we can
conclude from our previous work that given certain sequences an, for example the
sequence of primes or of squares, we have [a„a] prime infinitely often for almost
all a. Of course, our proofs actually dealt with [a„a], but one merely needs to
change
(p-^in) p + \jf{n)\ to (p_ p+\l/(n)\
\ an a, ) °W an )
in all the proofs to recover results for [an<x]. Our path has now come alongside
one of the central themes of analytic number theory, namely to prove that certain
'natural' sequences contain infinitely many primes. Examples of 'natural'
sequences include 2P — 1 (Mersenne numbers), n2 +1, p+2 (the prime twin problem),
etc. The sequences we consider here will be [a0-], [a£], and [<xan], where an is an
increasing sequence of positive reals. Indeed, we shall be able to characterize which
sequences an lead to infinitely many primes in each of the three cases. We are thus
able to establish 'almost all' analogues of results which to date have been
intractable in their original formulation. For example, we shall show that [ap] is
infinitely often prime for almost all a>l. The exceptional set here does not just
contain the integers, for we shall prove in Chapter 10 that it has Hausdorff
dimension 1.
The first problem of this type to be investigated from a metrical perspective
was the so-called Piatetski-Shapiro sequence [«"]. Piatetski-Shapiro [217] proved
that this sequence is infinitely often prime for all a with l^a^ 12/11 (of course,
the result is trivial for 0<a^l). Since his work, the range has gradually been
216 The integer parts of sequences
increased to 13/11 [149]. Deshouillers [72] proved that [na] is infinitely often prime
for almost all a > 1, and Balog [22] generalized his proof to give the correct lower
bound (at least infinitely often) for the number of primes, even when [na] is
replaced by [pa] (see Theorem 8.1 below). Their argument, which we reproduce
in Section 8.2, is basically the one we presented in Section 2.2 (although it was
not capable of giving any results in that context). It is successful here, since we
can give an upper bound to the number of solutions to
[na] prime, n^N, or [pa] prime, p^N
uniformly in a belonging to any interval not containing an integer. Here the
analytic quality of the sequence na is vital, leading to non-trivial bounds for the
exponential sums
which in turn enables us to estimate the remainder terms in the sieve. Balog
required a little more, namely estimates for
leGH
in order to apply a two-dimensional sieve method. The principle for estimating
these sums is the same (see Lemma 8.1 below).
Leitmann and Wolke [178] proved that the asymptotic formula
N
|{[«"]prime: n^N}\
alogiV'
holds for almost all ae[l,2]. There are serious difficulties which have yet to be
overcome before the result can be generalized to consider [pa], or to increase the
range of a beyond 2. Nevertheless, it has been shown [129] that the restriction
[na] to prime values could be replaced with other options, as shown in Theorem
8.2 below.
Theorem 8.1 Write
n(a,N) = \{[rix] prime: n^N}\,
7r*(a,A0 = |{[/?a] prime:/>^ A/}|.
Then, for almost all cc > 1, we have
om(<x,N)\ogN
hmsup — ^ 1
N-oo N
Introduction and statement of results 217
and
a7r*(a,iV)log2^^1
hmsup — ^ 1.
AT-oo N
Theorem 8.2 Let $0 be an infinite set of positive integers and write
jtf(a,N) = \{[na)esf: n^N}\, st(N) = s?(\,N).
Suppose that for some (re(0,1] we have s#(N)»Na. Then, for almost all
a e (1,2/(2 — c)), we have
st{x,N) = y X a~1+y + OC¥ll2(N)\ (8.1.1)
where y = a and, for any € > 0,
V(N)= X <s/(a)a~1+y+t
ae.w
Remark If s£ is the set of primes, then a can be taken arbitrarily close to 1, which
establishes the asymptotic formula for n(<x,N) for almost all a e (1,2). In [129] an
attempt is made to generalize this result to n*(a,N), but this requires a doubly
metric formulation of the problem, as a formula is obtained for almost all
(a,j5)e[l,2]2for
I 1,
[n*] = p
with p and q both denoting primes.
In the following results, first proved in [127], whenever we write log x we mean
max(l,logx).
Theorem 8.3. Let an be an increasing sequence of positive reals. Then [ancc] is
infinitely often prime for almost all positive a or for only a set of measure zero
according to whether the sum
oo 1 / \-l
„=ilog0„
diverges or converges, respectively.
Remark An equivalent condition for Theorem 8.3, which we shall use in the
proof, is the existence or not of a subsequence bn of the an such that b„+1^b„+\,
218 The integer parts of sequences
and with the following series divergent:
oo i
T —— •
n=llOg^„
Theorem 8.4 Let an be a lacunary sequence. Then, for almost all a > 0, the number
of primes having the form p= [ccan], n^N, is given by
*¥(N) + 0(¥(JV)1/2(log ¥(A03/2 +€)),
where
N j
WO=I;
„=iloga„
and € > 0.
Remark By Theorem 8.3 there are 0(1) solutions for almost all a when *P(oo)
converges, so we shall assume *F(./v")-»oo as h-»oo in the proof. Of course, *¥(N)
is the 'expected' number of primes of the given form, since the probability for a
number m to be prime is (logm)"1. Hence the probability that [<xan] is prime is
(log(aan))"1=(logan)"1+ 0((log<2„)"2). By analogy with our definitions in
Chapter 1, given an integer b ^ 2 we shall call a number a prime-normal to base b if
^T^\{[^bn]=p:n^N}\
N-+<x> lOgiV
exists and equals (log&)_1. That is, we truncate the base-6 expansion of a at the
wth place after the 'decimal' point, remove the point, and then obtain =(logiV7
log b) primes for n^N. If a number a is prime-normal to every base 6^2, we
shall call it absolutely prime-normal. By Theorem 8.4, since the union of a
countable collection of sets of measure zero itself has measure zero, almost all
positive reals are absolutely prime-normal.
Theorem 8.5 Let an be a strictly increasing sequence of positive reals with an->oo
as«->oo. Write
\an-am\<a\-0
Then:
(i) if Z(/?) converges for all j5>0, then [a^] is only finitely often prime for
almost all positive a;
(ii) if £(/?) diverges for all j5>0, then [a*] is infinitely often prime for almost
all positive a;
Introduction and statement of results 219
(hi) if neither (i) nor (ii) holds, then let
y = sup{jS: Z(jS)<oo}.
{Note that 2(j5) is non-decreasing function of /?.) Then [a*] is only finitely
often prime for almost all cce(0,y), and infinitely of ten prime for almost all
ae(y,oo).
Remark It follows that if an is any strictly increasing sequence with a„-»oo, and
an<nA for some A, then [d£] is infinitely often prime for almost all a>0. As with
Theorem 8.3, we remark that instead of discussing E(/?) we could deal with the
existence (or not) of subsequences bn such that \bn+1—bn\>bjl~fi and
oo
1
diverges. The proof will be given for this alternative equivalent formulaion. We
leave it to the reader to verify that the analogue of Theorem 8.4 holds for the
sequence [a*] when an is a lacunary sequence.
Theorem 8.6 Let an be a strictly increasing sequence of positive reals with an-+oo
as w->-oo. Write (for /?>1)
n=1un \ m^n ,
Then:
(i) */£*(/?) converges for all j5>l, then [cca-] is only finitely often prime for
almost all a > 1;
(ii) //"£*(/?) diverges for all j5> 1, then [a0-] is infinitely of ten prime for almost
all a > 1;
(iii) if neither (i) nor (ii) holds, then let
y = sup{0>l: Z*(0)<oo}.
(Note that 2*(j5) is a non-decreasing function of /?.) Then [a0-] is only finitely
often prime for almost all ae(l,y), and infinitely often prime for almost all
a e(y, oo).
Theorem 8.7 Let an be an increasing sequence of positive reals such that
an+1—an^c for some c>0. Write
FN(a) = \{[af']=p:n^N}\9 <!>(#)= £ A.
Suppose that <D(JV)-»-oo as 7V->oo. 77zew, /<?r almost all col we have
FN(a)=-J—<D(A^) + 0(<D(iV)1/2(log<D(^))3/2+€)
log a
/or any € > 0.
220 The integer parts of sequences
Remark This result gives an asymptotic formula for the two cases of greatest
interest: [an] and [ap]. Although we shall consider the dimension of the
exceptional set in Chapter 10, we note here that in the case of the sequence [an] it is
possible to specify some members of the exceptional set, namely certain PV
numbers (compare our discussion in Chapter 5 on the distribution of {a"}). Indeed,
Professor M.N. Huxley pointed out to R.C. Baker and the author that probably
the simplest such example is the number c = (21/3 —l)"1. To see this, note that
[cn] = 2or3(mod6),
and so the only prime value taken by [cn] is [c1 ] = 3. Of course, if we set a = c2,
then [an] is never prime. We remark that Mills [193] has shown that there exist
values A such that [Ay] is always prime. By the Borel-Cantelli lemma, the set of
such A has measure zero, of course.
8.2 PROOF OF THEOREM 8.1
In this section we shall give a complete proof of Theorem 8.1, but we shall also
prove Lemma 8.3, which will be crucial for Theorems 8.3-8.7. We begin with some
classical bounds for exponential sums.
Lemma 8.1 Let f(x) be real and have continuous derivatives up to the kth order,
where k^2. Let S^\fk\x)\^hS. Put rj = 22~\ T = (2fc-2)_1. Then
2N
I e(/(«))
n = N
In particular, for k = 2we have
«/zW^t+iV1-^-t. (8.2.1)
IN
I e(/(«))
n = N
«hNSll2 + d-112. (8.2.2)
Proof See Theorems 5.9, 5.11, and 5.13 of [263]. These bounds are due to van
der Corput, and the implied constants are absolute. □
To apply the above lemma to functions like na we shall need to avoid integral
values of a (which is hardly surprising!). We now couple Lemma 8.1 with standard
sieve estimates to establish the following.
Lemma 8.2 Suppose a>0, and [a,b]nZ = 0 Then, for cce [a,b],
n(a,N)<c-^- (8.2.3)
log AT
and
nHoLyN)<<:JL. (g.2.4)
\ogzN
In both the above inequalities the implied constants depend at most on a and b.
Proof of Theorem 8.1 221
Proof If b<\, then (8.2.3) and (8.2.4) follow from Lemma 6.4 (the Brun-
Titchmarsh inequality). For example, setting y = 1/a,
7i*(a,A0= £ I 1
«<N"«'<p<(«+l)T
3^((l+<T1)y-l)
4*
^J^log(^((l + <rV-l)
yqy~l 24Na
after some rough upper bounds are used. Henceforth we may therefore assume
that a > 1. To obtain (8.2.3) we shall need to apply a 1-dimensional sieve, whereas
(8.2.4) requires the application of a 2-dimensional sieve. We shall establish (8.2.4)
only: the proof of (8.2.3) is similar, but simpler. Let
jtf = {mn: 0^na—m<\, n^N},
s#d={res#: r=0 (modJ)},
S{fg)= I l-£,
m,n J y
mg = [(nfy]
n*N/f
p\d\ LPJ f\de\dlf J
We have
K.I=I I i-II*/) I 1
e\d (n,d) = e e\d f\d/e n^N/fe
mnes/ (dm/e) = [(nfeY]
m = 0(modd/e)
=Z I Kf)S(efd/e) + Z I rtf>Tf
e\d f\d/e e\d f\d/e aJ
=1 I Kf)S(efd/e) + ^^.
e\d f\d/e a
Thus
Rd = l I Kf)S{ef,d/e).
e\d f\d/e
222
The integer parts of sequences
Using the 2-dimensional sieve as in Theorem 5.2 of [108] (compare Lemma 7.2)
to sieve «s/, we thereby obtain
7r*(a,A0«Arn(l-—)+ I d' % \S(ef,d/e)\ (8.2.5)
p^z\ P J d^z fid
M2(z)=l e\d/f
for any €>0 and any z>\. In applying the sieve, we have noted that co(d) is
multiplicative and satisfies
I ^!£^=21og(z/lv)+o(i).
w^p<z P
We pick z=Nplk with p = 2"fc-10, k = [b]+2. This gives a bound «N/\og2N for
the first term in (8.2.5) (and with an implied constant depending only on b). The
second term may be estimated as
*2€ Z \S{f,g)\.
We note that if g = l, then \S(f,g)\ = 0(l), since there will be exactly one solution
in m to the equation mg = [(«/)"] for each pair n,f. If g^2, this problem reduces
to counting the number of solutions to
{(nf)*/g}<g-K
Thus S(f,g) can be estimated via Lemma 5.1. We obtain
|S(/,9)i<7E+o(i)+(H)i
I e|
n*N/f
'Knfy
9
We split the sum over n into blocks of the form n~B. We then need to estimate
sums of the form
I e(fri))
where
«*) =
lfaxa
9
so
Hence
fW(jc) = a(a-l)--(a-fc + l)
If*
a-k
0
//v
Proof of Theorem 8.1 223
where // and v depend only on a and b (in terms of their size and the sizes of || a \\
and || 61|; it is at this point that we require ||<z|| ||&||>0). We now pick € = p100
and put L = gN10e, which gives a bound after applying Lemma 8.1,
jyl-lOi;
\S(fg)\< —r-,
f9
and hence (8.2.4) follows. □
Although not needed for the proof of Theorem 8.1, we prove the following
variant of Lemma 8.2 which will be vital later.
Lemma 8.3 Let R^2, v^-\be given, and suppose that
Then the number of solutions in primes r, s to
\rd-s\<2, r~R
(where, as usual, r~R indicates R^r<2R) is at most
KR
log2/?'
where K is an absolute constant.
Proof We cannot appeal to Lemma 6.4 here, since 0 may be too close to 1. We
follow the previous proof with k = 2. The bound for the exponential sum arising
is now
with
lfSRe-2
<5 = 0(l-0)
9
We have S^lfiRgy1 ^R^n+amp-i^ and so the term ^51/2 poses no difficulties.
On the other hand, 0-l^iT3/4, and so
(5>JR-7/4^-1>JR-7/4-(1/2)^JR-15/8.
The proof now follows as before. □
We now give a quantitative version of the argument presented in Section 2.2.
Lemma 8.4 Let A and B be reals with B>A. Let FN(ct) be a non-negative-valued
function ofN (an integer) and a (a real variable), and GN a function of N such that:
(i) GN->oo asN^oo; (8.2.6)
224
The integer parts of sequences
(ii) for all sufficiently large N and for all a, b with A^a<b^B we have
(8.2.7)
limsup | —^—d<x^(b —a);
N^co ]a GN
(iii) there is a positive constant K such that, for all ae[A,B],
FN(<x)^KGN.
Then, for almost all a e [A, B], we have
(8.2.8)
limsup
N-oo G
E*w>lm
(8.2.9)
N
Proof Write
HN(oc) =
FN(oc)
'N
Suppose
limsup//N(a)<l
N-oo
[A, B] with positive
on a set with positive measure. Thus there exists a set s&
measure and a constant c<\ with
limsup HN(ot) ^ c for a 6 si.
N-oo
By the Lebesgue density theorem, for each € >0 there is an interval [a, b] <z [A, B],
b<a + l,withl(@)>(l-£)(b-a), where @ = jtfn[a,b].Pick£ = (l-c)(b-a)/2K.
We then have
rt
HN(a.)dcc =
HN(ot.)dcc+ I //N(a)da
3» J[a,b]\&
HN(<x)da + K€,
3t
by (8.2.8). Thus
limsup
N-oo
rb
HN(<x)d<x^c(b-a) + Ke = i(l + c)(b-a)<b-a.
This contradicts (8.2.7) and so establishes (8.2.9). □
Proof of Theorem 8.1 In view of Lemmas 8.2 and 8.4, it suffices to prove that
,ba7r(a,A01ogiV
N
da^b-a+o(l)
(8.2.10)
Proof of Theorem 8.1
225
and
"fc/v<n-*
a7r*(a,AT)log27V
~N
doi^b-a + o{\)
(8.2.11)
We shall establish (8.2.11); (8.2.10) follows similarly. We use the standard notation
Mn) for the von Mangoldt function defined by
[0 otherwise.
We shall then use the prime number theorem in the form
JV
YJA(n)=N+o(N)
n=l
and Mertens' theorem in the form
£^-logJV+0(I).
n=l
n
We have
^a7r*(a,A01og2iv"
N
alog2#
da
la " p^N
Z Ida
>
rb 1 /log2N\
[n*] = m
1
= -£ A(«)A(mU|
logw log(m+l)
log«' log;
+ 1)1 r h,\^n{^g2N\
= \_ m y Mm) ( /log2AT [ yl A
^nlog«j£^ m \Nm J
41 ^(0-.)log« + O(l))+o(^ + ^^
= ^rl A(n) + o(-\\=b-a + o(\),
N hTn Vlog^/
as required to complete the proof.
□
226 The integer parts of sequences
8.3 PROOF OF THEOREM 8.2
In this section we shall write y = oc~1, so y e[-J, 1]. Let
'1 if a = [na] for some n,
Then
10 otherwise.
g(y,a) = (a + \y-ay + ({-(a + \y}-{-ay})
=ya?-1+0(ay-2) + ({-(a + \y}-{-ay})
= ya?-1 + 0(ay-2)- £ _L(e(fc,')-e0(fl + iy))
0<\h\^H^7lin
-(-('•si)-K'-™i
for any H, using the familiar truncated Fourier series for {0}.
Hence
Dl
g(y,a)=ya?-1+OW-2)-E1(a,y) + E2(",y),
where
E1{a,y) =
and
X t(h{a + ty){a + ty-xydt
0 0s£|/iKff
^,y)=o(min(l,^) + min(l,^
Dl.
To prove Theorem 8.2, we first restrict our attention to an interval
[v~ \u~1] c(l,2), where v = u+^£. We may then obtain our result by considering
almost all y e [w, t?]. We now choose
^=^(a) = [^(«)1/4a^] + 2, where 0=^.
It will become apparent later that this is the optimum choice for H to balance the
two errors E^y) and E2(a,y). Clearly,
aesJ
is our main term, while
£ 0(^-2)=0(l).
Proof of Theorem 8.2
227
Henceforth the variables a and b will both tacitly be assumed to belong to s#.
We shall show that, for any s > r ^ 1, we have
X \E2{a,y)\dy« £ //(fl)-1log(2a) = 0>1(r,5), say, (8.3.1)
u r^a<s
r^a^s
and
Z E\(a>i)
r^a<s
dy« £ H(a)2\og3(H(a))a-^
r^a<s
= <D2(r,5), say.
(8.3.2)
By the argument used to prove Theorem 4.1 it follows that, for almost all y e [w, t?],
we have
I 0(7,a) = rl ay-1+0(01(l,7 + D1+(1/4)€) + 0«>2(l,7 + l)(1/2)+(1/4,€). (8.3.3)
By Cauchy's inequality,
o^+D2^ x //(fl)2iog4afl-4M x
H{a)-*a
~*nW
a<«<y / M^a^y 1°S ^
J^(fl)
-1'
« X ^(^log4^"2' I . 2
Since the second sum in (8.3.4) converges, the error term in (8.3.3) is
(8.3.4)
\(l/2) + (l/4)€\ // \l/2\
0(( X ^(a)ll2\og4aa-2^) ) = 0[[ Z ^(a)172^"1^] )
,U««jf
as required to complete the proof of Theorem 8.2.
To obtain (8.3.1) we note that if A^n7^A + d, then y is restricted to an
interval of length
\og(l + 6/A) S
log n A log n
Thus
min(tf, ||nl-*)<!?<< I
n"-l^m^n' + im^°Bn fc=l
«log(2#)«log(2«),
i l+[log2H]
X 2-fcmin(#,2k)
and this gives (8.3.1).
228
The integer parts of sequences
We begin our demonstration of (8.3.2) by splitting up the range of summation
over h into blocks, which gives
fv
oo
n
Z Z Z eihia + tyXa + ty-'ydt
fc = 0 |/i|~2* r^asgs JO
H(a)> \h\
dy
n
2 oo
rv
2->k
b k = 0
m
Z Z ^{a+ty^a+ty-1
0\h\~2" r^a<s
H(a)>\h\
dtdy (8.3.5)
by three applications of Cauchy's inequality.
Henceforth we shall tacitly assume that the integer variables a and b are
restricted by H(a), H(b)^\h\. We have
12
Z eWa+tMa + ty-1
r^a<s
dy
rv
= Z (a+ty-^b+ty-^hua+ty-ib+tyyidy
r^a,b<s Ju
« Z (a+ty-1(b + t)v-1mm(\,
1
^a,b<s \ \h\\(a + ty\og(a+t)-(b + ty\og(b + t)l
using Lemma 5.3 with integration by parts,
« Z fl~2'6~2'min(l,,y, L * xl
^ \'\h\(b-a)loga,
r^b^a<s
1
< I a~4p+^h I
l
r^a<s
\h\r<b<a<s(a-b)\oga
1
(8.3.6)
r<a<s
The right-hand side of (8.3.5) is thus
<f(A: + l)22' W £ a-w + i; I a"2'
fc = 0
|fi|-2*\r^a<s
Mr<a<:
« Z (a~^H(a)2log2(H(a)) +a~2pH(a)\og3(Ha))
r^a<s
(recalling that a was restricted by H(a)^\h\)
« Z (a~*i>H(a)2log3(H(a)),
(8.3.7)
r<a<s
as desired. To obtain (8.3.7) we noted that H(a)a~2p^l follows from
a ^2(2 — c)"1. We have thus obtained (8.3.2) and so completed the proof of
Theorem 8.2. □
Proof of Theorem 8.3
229
The reader should note that it was the diagonal terms (a = b) which gave the
most significant contribution to (8.3.6). If s$ were to depend on a in the way
necessary to obtain a formula almost everywhere for 7i*(a,A0, the analogue of
(8.3.1) would be easily obtained, and the diagonal terms would cause no difficulties
in getting to (8.3.6). It is the non-diagonal terms which would now become very
difficult to estimate.
8.4 PROOF OF THEOREM 8.3
We first need two further lemmas which are applications of the two-dimensional
sieve.
Lemma 8.5 Let R^Z^R1/2^2, 3^A<R1/3. Then the number of solutions in
primes r, s to
\r-s-A\<2, R^r<R + Z
is at most
KZ
n
lotR\A-„\<2(p(n)'
Proof This follows from Corollary 2.4.1 in [108] with k=a = \.
Lemma 8.6 Let R^2, JT1/5^<5^1, ir3/4^0^1-6JTx.
Suppose also that
□
a
e--
-2 r>-l
^S~2R
>q>S~\
Then the number of solutions in primes r, s to
|0r-j|<2, R^r^R + Rd
(8.4.1)
(8.4.2)
is at most
KRS
log2*'
(8.4.3)
Remark The exponents 1/5, 3/4 in the hypothesis are not significant: we wanted
to avoid an unnecessary arbitrary €. A condition like (8.4.1) is necessary, since if
it is violated, it is possible for the number of solutions to rise by a factor of order
of size loglog<5_1.
Proof By Theorem 2.1 there exist coprime integers a, q with 1 ^ a ^ q ^ *3/4 and
e-a-
q
l
qR3/*
230 The integer parts of sequences
We shall suppose that 6>a/q in the following, for clarity, and write X = 0—alq.
We split the range for r into blocks [H,H+Z], where Z = qR1,2°. Then
ra
6r-s= s+m + 0(R-1110).
We therefore need to bound the number of solutions to
<3 Wwh A = [Rl+\],
ra
s+A
q
(Note that A=0 for all sufficiently large R unless q<2R1,A: for the case A=0
we could appeal to Lemma 7.2, but the proof described below works for all
possible A.) That is, we count the number of primes r, s with
ra = q(s-A) + b, re[H,H+Z], \b\<3q.
We first dispose of the case a = q = \. In this case we obtain (8.4.3) from Lemma
8.5, since the inequality 0^ 1 — 6R~1 ensures that we do not obtain solutions with
r=s.
We can now suppose that a < q. For each b the solutions have the form
faa-\\
r=ab + nq, s=A + na + bl 1, (8.4.4)
where aa=\ (mod#). By Theorem 2.3 in [108] the number of solutions is
aq qA—b Z
« —
q>{aq) <p(qA -b)q log2/?'
We here use the convention that <p(ri)=—(p(—ri) for n<0, so that (qA—b)/
q>(qA—b) is always positive. Since there are no solutions to (8.4.4) unless
(b, q) = (qA—b,a) = \, our bound is
Z aq ^ qA + b
« x
q \og2R q>(aq) {b£3q q>(qA + b)
(b,q)=l=(qA + b,a)
Z aq „ n Z
« —. T= — 7 > — «
qlog2Rq>(aq) (n^=1 <p(n) log2*
\n-Aq\<3q
by (7.2.8), provided that A is not so large that log A behaves like a power of q.
Summing over the blocks, we thus obtain (8.4.3) unless q is very small.
If q is small, we must employ (8.4.1) and make use of the averaging over A.
Write
A0 = [XR+H A^Ao-3, A2 = [XR(\ + S)]+4.
Proof of Theorem 8.3
231
We may now suppose that A0^q2. If 6—a/q<3~2R~1, then A0<S~2^q2 by
(8.4.1). We can thus assume that RX>5~2 and so S(RX)1/2>\. It follows that A
takes on >RSX — \>(RX)1/2 — X^Aq12 — 2 values. We therefore obtain a bound
by Lemma 7.4.
«
aq
o^Rd/zqfoS R<p(aq)
1*1 <3«
(b,q)=l=(qA + b,a)
A = [(R + nZ)X+ 1/2]
V(qA + b)
«
aq 1
«
tf<5
«
#log Rq>(aq) ZX („,aq)=i q>(n) log2*
qA1^n^A2q
Proof of Theorem 8.3 We begin by supposing that no subsequence exists with
bn+2>bn + \ and with
oo
1
(8.4.5)
.-ilog^
divergent. Now pick a subsequence bn of the an by taking
bi=ax, bn = min{am: a^lb^^ + l} for«^2.
We note that for each ah with k^ah<k + X there is an n such that k^bn^ah.
Clearly we have bn+2>bn + X, and so (8.4.5) converges. Let c>0 be given, and
we shall consider a e X = (c, c +1). By the first Borel-Cantelli lemma it follows that
the expression [cnbn ] + d, nEN,0^d^c + 2, is prime only finitely often for almost
all a e X. To see this, note that
[<xbn] + d prime o cce
p—d p — d
L n
—1
n J
and
oo
n= 1 p,d
'p-d p-d + \
oo 1
-I I F" I
n=l 0s£ds$c+2 yn p
c^((p-d)/fc„)<c+l
1
00
1
^ + 3)Ir (7Z(bn(C + 1) + C + 2) - 7T(V))
oo
1
«
,-ilOg^
by the prime number theorem (or just Chebyshev's upper estimate for 7c(jc),
namely 7i(jc)«jc/logx). Now, if [ocah] is prime, it follows that [ocbn] + d is prime
for some n and d with 0 ^ d^c + 2. This completes the proof of Theorem 8.3 in
the convergence case.
232
The integer parts of sequences
Now we can suppose that an+1^an + l, and that
oo i
i
(8.4.6)
n=i^ogan
diverges. Let B>A>0 be given, and we consider oce/ = (c,c + S)<=[A,B]. Write
)n = [A,B]n{)\^
ln = ®nnf.
We then want to show that almost all a belong to infinitely many @n, and we shall
use Lemma 6.1 to establish this. We have
V Ka )= y <{c + d)an)-n{can) + 0{\)
n=l
n=l
a„
N \ f ( \
n=Joga„
by the prime number theorem in the form
log**,
n(x) =
+ o
x
log* Uog x
To apply Lemma 6.1 we must therefore demonstrate that
,2
1 ^m,n^N
...-ilogfl,,,
(1+0(1))
=KSS2(N)a + o(l)),
(8.4.7)
say, where K is independent of S and c (although it can depend on A and B) but
the implied constant in o(\) may depend on S. As in Chapter 7, we stress that this
is not a tautological use of implied constants: we have two limiting processes at
work and y=o(l) means y-*Q as N-*co, even if for fixed N, j>->oo as <5-^0. For
the rest of this chapter, K denotes such a constant, not necessarily the same at
each occurrence. We may therefore write K+K2<K for example. Now the
contribution from the terms m=n is 0(S(N)), so we need only discuss the sum
with m<n. The intersection of 0&n and 0$m is then made up of the union of the
intersections of pairs of intervals thus:
P />+!'
an an
n
q q + \
Pm am
(8.4.8)
where
0„
a
m
i p q
a,
m
an am
(8.4.9)
Proof of Theorem 8.4 233
We note that the measure of (8.4.8) is ^a'1. We rewrite (8.4.9) as
\dp-q\^l, where 0=^. (8.4.10)
We bound the contribution from terms with n <,5~5 trivially. Suppose now that
(canyll2^d. We have 6 = l-(an-am)a;1^l-6(ca„y\ provided that
an—am ^ 6c'1. Hence, by Lemma 8.6, the number of solutions to (8.4.10) is
Kda„ Kba„
< JL < »
\og2an (\ogan)(\ogam)
assuming that (8.4.1) holds. The terms for which (8.4.1) fails are included below
as the final sum in (8.4.11).
On the other hand, if 6<{can)~112, let T=d~1 and consider
\p-qT\^T.
For each q the number of solutions in p will be no more than 3(27+ l)log~ 1T by
the Brun-Titchmarsh inequality (Lemma 6.4). Summing over the Sam\og~1am
values of q then gives a bound
K5am
(\ogan)(\ogam)
Hence
^bam\o%-xamT\o%-xT<
Z ¥„nlm)
1 ^m,n^N
^6-10 + O(S(N)) + KSS2(<N)+ Z K@n^@m)
1 ^m,n^N
\an-am\<6/c
+ Z I Z %nlj. (8.4.11)
q^S~1a=l ls£m,nsSN_
\qam-ban\<qS 2
Since \an-am\^l for m^n, the fourth term on the right of (8.4.11) is 0(S(N)\
while the final sum is 0(S(N)) with an implied constant whose dependence on 3
is like <5~4. Hence we obtain
Z U&« n # J < 0(S(N)) + *<5S2(A0
1 ^m9n^N
= KdS2(N)(\ + o(l)),
as required. The proof of Theorem 8.3 now follows from Lemma 6.1. □
8.5 PROOF OF THEOREM 8.4
We now require more accurate estimates for 2(JBnJm). We consider
X = [A,A + \] for some A>0. First we note that the intervals in (8.4.8) intersect
234 The integer parts of sequences
only if
-\<p-q—<—.
If am^aj,13, by Huxley's result on primes in short intervals [142] the number of
solutions in p for each q is
«■ W '
aJogan\ \\ogan/
Hence
log an log
l0gflnl0gam \M^m^n^Nlogfln/
since
oo i
z
Bf i log2tf„
converges by the lacunarity of the sequence an.
On the other hand, using the results of Section 8.4,
£ M.n*J<K I . ' +o( I j-i-
M
\M<m<n<NlOg^n/
again using the fact that since an is a lacunary sequence,
v. 1
m
tn X°%an
is bounded independently of n. Writing fk(x) for the characteristic function of ^k,
then
[A+1( Z (/*(*)-pMYd*^ a: x -i-.
Thus Theorem 8.4 follows from Lemma 1.5. □
8.6 PROOF OF THEOREM 8.5
We now take A,B,C^ as in Section 8.4, and write
Proof of Theorem 8.5
235
We have
Wm)= I
l + 0(p~l)
loglog(flc„+*) - loglog«)+o{\)
(by Mertens' prime number theorem)
_log(l + <5/c)+<?(!)_
log**
c\oga\ \A
We assume first that an+x — an>a\ ~A and that (8.4.6) diverges. We further assume
that A^\. If A<\, it is necessary to modify some parts of the argument, since
this alters which is the smaller prime. In order to apply Lemma 6.1 to prove that
there are infinitely many prime values of [a*] for almost all oceX, we must
establish that
£ Wmnam) <KSS(N)2 + 0(S(N))-
1 ^m,n^N
As before, we can restrict ourselves to the case m<n. We have
log/? log(/7 + l)~| riog# log(# + l)~|\ log(l+ !//>)
M
log a/ log
while the intersection is empty unless
log/? log?
\ogan \oga„
+ 1)1 Hogg log(g + l)1\ log(l + l
«. J Ll0gflm' lOg^m J/ lOg^n
<
<llogam
We rewrite (8.6.2) as
where
\log(pe)-\ogq\<q \
6 =
log**
(8.6.1)
(8.6.2)
(8.6.3)
We note that (8.6.3) gives crudely 2/3<pe/q<3/2 (assuming q>3). By the
mean-value theorem,
\og(pd)-\ogq =
pe-q
h '
with
pe<h<q or q <h<pe
236 The integer parts of sequences
Thus it suffices to consider the inequality
\p6-q\<\. (8.6.4)
We first divide up the range of p into blocks of the form [R,2R). If
i<0<l-/T3/4, the desired bound follows from Lemma 8.3. If 0>l-*~3/4, we
split the range for p into the smaller blocks [Z, Z + Y\ R ^ Z < 2R, with Y=R112.
It then follows from several applications of the mean-value theorem (alternatively,
write p=Z+y, pe=exp(d\og(Z+y)) and use the power series for log(l + Jc) and
exp(jc)) that we have
P6=P + (P6-P)
=p+(6-\)(p\ogp + 0((0 -1)/ log2/;))
=p + (0 - 1)Z log Z + O((0 - 1)2Z* log2Z + (0 - \)Y log Z))
=Jp+(0-l)ZlogZ + OCR-1/4log2JR).
We thus wish to count solutions to
\p-q-F\<2,
and from Lemma 8.5 we obtain a bound
KY y n
log2* ,Fij<2 <?(")'
provided that f>3 (note that (l-0)ZlogZ = 0CR1/4logtf)<JR1/3 for all large R).
Summing over the values of Z between R and 2R then gives a bound
KY y y n < KY 1 |, n
Wr\ |F4J<2 <p(«) ^ log^rd-^log.R^ <p(n)
F=(i-e)z\ogz
(here Q = 2(\-B)R\og(2R) + 2)
KR
^
log2/?
as required, using Lemma 2.5.
Now, if F<3, then (1 — 6)R\o%R<A, which gives
6V am ) RlogR R
Since \an—am\ > a\ ~A, this happens for a bounded number of values m for each n.
Finally, if 6<j, then we can replace (8.6.4) by
\p-qT\<4TR1-0, where T=0_1.
The result then follows by Lemma 6.4.
Proof of Theorem 8.6 237
Combining the results for all the cases gives (8.6.1) as desired to prove the
theorem when £(/?) diverges. To establish the result for the convergence case we
leave the reader to modify the analogous result in Section 8.4. Suffice it to say
that the subseqeunce bn is now chosen so that no more than one member lies
between nllp and (« +l)llp for n = 1,2,... □
8.7 PROOF OF THEOREM 8.6
We shall be brief with the details, as the structure of the argument is as before.
We suppose (c,c + d)<=[A,B], 1<A<B, an—am>a~A. Write
Then
<f"-l<p<(c + d)a"
K@n)= Z P°-((\+p-ly—\)+0{c->)
> £ c((l+/7-ir,_l) + 0(c-p)
ca"-l<p<(c + d)a"
>^- Z p-l+(Kc->)
for all large an,
c
= -- (loglog(c + d)a" - loglog ca-) + 0(a~ 2 + c ~ «•)
(by Mertens' prime number theorem)
AS
>
3a„B\ogB'
Here we have used the mean-value theorem and assumed that an is sufficiently
large. Thus
n=l n=l
Now we have intervals from Mn and @m intersecting only if (we assume m<n
as before)
\pd-q\<2, where 6=^.
Also, we have Mnc\$m made up of intervals
[^-,(/7 + ir-]n[^,(^ + ir»],
238 The integer parts of sequences
which have length
B
Consider the contribution from p~R, and treat first the case 1 — R~3/4"^d>j,
where Lemma 8.3 can be applied. This gives a contribution to the overlap
estimates (omitting summation conditions for the sake of clarity)
an\og2R
771, 71 771, 71
as required. In the above we noted that, since
ca*-\^R^(c + d)\
we have
R
where K depends only on A and B.
For 6<j we note that intervals intersect only if
qx-l^p^(q + \)\ where t = 0_1.
We can now apply the Brun-Titchmarsh inequality (Lemma 6.4) to obtain
m<n^N \n<N /
0<l/2
as required.
It now only remains to consider 0>1 — R~314. We can argue as in the
corresponding case in Section 8.6. When F<3, we get (1 — 6)R\ogR<4, which
gives (an—am) R log £ < 4, where c ^ £ < c + <5, and so an—am< (K/R) < Ka~ A. This
occurs only a bounded number of times for each n, in view of our assumption on
the sequence. The proof is completed with an appeal to Lemma 6.1. □
In the case of Z*(/?) convergent, pick a subsequence bn of the an with no more
than one member from each of the intervals
logw log(w + l)
|_loi/r log/5 J
w=l,2,
8.8 PROOF OF THEOREM 8.7
We sketch how the ideas of Sections 8.5 and 8.7 may be combined to prove
Theorem 8.7. Attention is first restricted to the interval [A,B], where A>\.
Proof of Theorem 8.7
239
Lemma 1.5 is applied with
/*(*) =
flog* if x°k=p
0 otherwise,
fk = <Pk = ^k1-
Write
*/*) =
(logx)-'dx,
s
where £ is any Lebesgue measurable subset of [l,oo). Then
-4 \u^k<v / u^m,n<v
-20(M,r) X ^.Htf-^tei;),
u^n<f
where
<D(w,iO = X fk-
As in Section 8.7, we can give an upper bound for X2(^mn^n). When 3am<an,
however, we can given an asymptotic formula with error term Oia'1), using
Huxley's result as in Section 8.5 (the log2Jc factor causes no problem). The error
term thus has the form <t>(N) as required when 3am<an. The upper bound used
for an < 3am also leads to a term with the required size. The correct formula for
kx{08H) is easily obtained, since if we write
X(*) =
\0 if [jc] is composite
1 if [jc] is prime,
then
rB
>n(@n) =
logJC%(jca")dx=
'B*
logyx(y)'
,!/«.-!
at
dy
= I
Aa"<p<B°'
log/7/7
!/«„"!
a:
(l + OC/?-1))
CB-- l/am-l
a:
(l + Odog-^dj;
B-A
0.
(\ + 0(a;1))
The proof then quickly follows.
□
240 The integer parts of sequences
NOTES
The result that [ntx] is infinitely often prime for all a £ Z was first noticed either
by Heilbronn (see page 180 of [271]) or by Fogels (see [178]); neither of these
mathematicians published this result.
It would be of great interest to replace 'limsup' with 'lim' in Theorem 8.1, but
the only way we know how to do this requires overlap estimates. As can be seen,
the currently available estimates are not sufficiently good to establish asymptotic
formulae unless the sequence grows rapidly.
The exponential sum bounds we quote from Titchmarsh [263] may also be
found in Huxley's recent monograph [143].
It should be possible to extend all the results quoted to higher dimensions, for
example to make
foaiM*^]. •••>[«.,«*]
simultaneously prime. It easily follows from Theorem 8.3 that for almost all
aeIR*-1 there are infinitely many primes px such that each of the numbers
Pj=[Pj-i0cj_1] is prime (2^j^k), or one could take Theorem 8.5 and obtain
the same result for the numbersPj= [/>"_ J. Professor R.R. Hall asked the author
whether it was possible to show that infinitely often,
/>,[/>"], and [[/>«]«]
were all prime for almost all a > 1 (and the result extended further). This is proved
in [128] for almost all a>2. Technical problems arise when generalizing the proofs
in this chapter to this situation: results like Lemma 8.3 are needed with the
variables constrained to short intervals, and this constrains the size of 6 (especially
for a near to 1). In fact, a > 20/19 is required just to make the 'main term' (that
is, ZpXi^p)) correct, using results for primes in almost all short intervals.
One could consider other generalizations, such as making [anoc] and [anoc2]
simultaneously prime. The most natural result in this area would be to make px
and each of
Pj=[<xPj-i]> Kj^k
simultaneously prime. A related question regarding solutions to
p=[n"iq = [n*2>]=- = [nf]
is discussed in [129].
Of course, the results we proved for prime values can usually be modified to
give other values which result from a sifting process (such as sums of two squares),
with suitable changes to the conditions. For example, (log«n)_1 becomes
(logtf„)~1/2 in Theorem 8.3 when dealing with sums of two squares.
It is frustrating that results like Theorem 8.2 are valid only for ae[l,2]
(at most). Presumably the asymptotic formula still holds for almost all a>2.
9
Diophantine approximation on manifolds
Mahler's classification of transcendental numbers. A simple result on
linear forms. Sprindzuk's theorem. Schmidt's theorem for smooth curves
in U2. Essential and inessential domains.
9.1 INTRODUCTION
In Chapters 3 and 4 when we considered simultaneous approximation, we were
interested in what was true for almost all ae!Rk. The methods presented there
could not deal with questions of what is true for almost all a on some submanifold
r of Uk with respect to the measure on T. For example, for almost all (txl, a2)e IR2
there are only finitely many solutions to
W^nWKn-1'2-*, KhHoz-1'2-
for any € > 0. This tells us nothing about the number of solutions to
\\cm\\<n-ll2-\ ||a2«||<«-1/2-€
for almost all ae IR. At this point we mention that, by a well-known 'transference
principle' (see Chapter 5 of [58]), the statements
'max ||a,.«||<«-1/k-€
has only finitely many solutions for aeIRk for all € > 0' and
<||a1«1+---+akrtk||<( maxl^-l ) (9.1.1)
has only finitely many solutions for ae IRk for all € > 0' are equivalent. Of course,
by Dirichlet's pigeon-hole principle, there are infinitely many solutions for both
statements if the € is removed (compare Theorem 2.1).
Great interest has been shown in formulation (9.1.1) of this problem
since Mahler's classification of transcendental numbers in 1932 [186]. For a
242 Diophantine approximation on manifolds
transcendental number 0 (real or complex), write
1 -log(|flk0fc+-+a10+ao|)
co(k, n) = — max —: ,
k log/z
where the maximum is over values of a0,...,ak with 1 ^ max \aj\ ^ h. Here h will
be called the height of the polynomial a0 + axx-\ \-akxk. For real 0 we could
replace
\ak6k+ ••• +^0+^01 with \\akdk+ — + ^011,
since the disappearance of the size of a0 is of no consequence: if |<z0|>|<Zj| for
l^j^k, and \ak(P-\ \-ax0+aQ\<\ (which it must be when taking the
maximum value of its reciprocal's logarithm for all h> |0|), then, for at least oney, we
have
as h->co. We write
cok = limsup co (k, h) and co = limsup cok.
h-*oo fc-»oo
It follows by Dirichlet's box principle that cok ^ 1 for each k if 9 is real, and
cok^l/2 if 0eC\R. If cok<oo for all k, we write v=oo. Otherwise we write
v = mm{k:cok = co}. Mahler's classification is then as follows:
0 is an S-number if 1/2^a>< oo, v= oo;
0 is a T-number if co = oo, v = oo;
0 is an A-number if co = oo, v < oo.
Henceforth we shall be concerned solely with real 0. Mahler conjectured that
almost all real numbers were S-numbers with supfccok = l. This was eventually
proved by Sprindzuk [252], after earlier authors had shown that for almost all 0
we have co2 = coz = 1 ([174], [272]). We shall give a refined version of his result due
to A. Baker [3] in Theorem 9.2. The value supk<wk for a given S-number is called
its type.
We first state a very simple result on problems of the type (9.1.1) to put our
other results in context. We write 0>k for the primitive integer points in Uk, that is,
&k = {neZk: GCD^ ,... ,«k) = l}.
Theorem 9.1 Let peR, \J/(n) (« e Zk) be a positive function of k integer variables,
\j/{ri) < 1/2, and, for a 6 IRfc, let jV(h,a,p) be the number of solutions to
11/11^+•••+/iJkaJk + )?||<^(ii), l^maxl^l^/z, (9.1.2)
and jV*(h,v.,fl) the number of solutions with ne^k.
Introduction
243
Then:
(i) if
X iM«)<oo,
«eZ*
Mew ^(/*, a, f$) is bounded for almost all a as /z -► oo;
(ii) (f
Z <M/I)=00>
Mew ^(/z, a, /?) w unbounded for almost all a as h^-cc; indeed,
jV*(h,ap) = T¥(h) + OV¥1>2(h)(\ogh)3l2+*), (9.1.3)
where
1 < max |n;| < h
for any € > 0.
Remark The reader should compare this result with the 'dual' results of
Theorems 3.6 and 3.8. In particular, we note that whereas a coprimeness condition
could be introduced implicitly in the proof of Theorem 3.8, we have had to include
one explicitly here.
Proof We need only consider ae [0, l)k. We use Lemma 1.2 in a standard way
to deal with case (i). We note that the set of a with
||fl1a1+"-+!!tat + 0||<^(i!)
for fixed ne Zk is measurable with respect to the product measure on IRfc. Suppose
that «j#0. Then, for any values of the txt (t^j),otj is restricted to a set consisting
of a finite number of intervals with total measure 2\p(n). The result then follows
by integrating over a, (t #/) and summing over n to produce a convergent series.
Clearly the first assertion of case (ii) follows from the second, and the reader
will recognize (9.1.3) as an application of Lemma 1.5. Expanding the characteristic
function, say %(«, jc), of the set {jc: || jc|| < \j/{ri)} as a Fourier series,
£cr(ii)e(rx),"
r
we have
/»
X(n,aLn + p)x(m,cun + p)daL
J[0,1)*
= J>r(/i)cs(m) e(r(*n + P)+s(<xm + p))d*. (9.1.4)
r,s J[0,1)*
244 Diophantine approximation on manifolds
Since (/i1}..., nk) = (ml9..., mk) = 1, it follows that ntj+smj is non-zero for at least
oney unless either m=n or both of r,s are zero. Hence the integral (9.1.4) is simply
c0(ri)c0(m)=4\l/(ri)ij/(m) if m^n, and 2ij/(n) if w=/i. Thus (9.1.3) follows from
Lemma 1.5. □
Now that we turn to consider the submanifold (ak,ak_1,...,a), we must
dispense with the generality of Theorem 9.1 and restrict ourselves to the case
\l/(n) = \l/(ma.Xj\nj\), with \j/ monotonic. Improving Sprindzuk's result (which had
i/^(/i) = (maxj|«j|)"k"€), A. Baker proved the following.
Theorem 9.2 Let k^\, and suppose that \j/Qi) is a monotonic decreasing function
for which
h=l
converges. Then there are only finitely many solutions to
||«1a+---+«fcak||<iAk(/z), l<max|/ij|</i (9.1.5)
j
for almost all cceU.
Remark Ideally, the inequality in (9.1.5) would have h1 ~k\j/(h) in place of \j/k(h),
but the reader will see in the proof how the inductive argument requires \l/k(h).
Bernik [32] has given a proof of the best possible result. Of course, Theorem 9.2
immediately answers Mahler's conjecture, to give the following.
Corollary Almost all real numbers are S-numbers of type 1.
The result of Theorem 9.2 was motivated, as we saw, by the desire to establish
Mahler's conjecture. However, it naturally raises the question as to the form of
submanifold required to establish analogous results. Submanifolds for which these
hold are called extremal. At the time of writing, the best results deal only with
curves in dimensions 2 or 3 ([239], [28]), although other types of result are known
(see Chapter 2 of [254]). We prove here the following result of W.M. Schmidt.
Theorem 9.3 Let T = (x(s),y(s)) be a curve in U2, with arclength s as a parameter
(a^s^b), where the third derivatives x(3)(s),y{3)(s) exist and are continuous, and
the curvature
k(s)=x'(s)y"(s)-x"(s)y'(s)
is non-zero for almost all se [a, b]. Then there are only finitely many solutions to
||«1jc(s) + «2.y(s)||</r2~€, l^maxlM;!^/*, (9.1.6)
j
for almost all s e [a,b].
Proof of Theorem 9.2 245
Remark R.C. Baker [15] replaced the term /z~2-€ with \j/2(h), where
h=l
converges, and very recently it has been shown that h'^xj/Qi) suffices [35].
The dual formulations of Theorems 9.2 and 9.3 have not received so much
attention. Thus, whereas it follows from the transference principle mentioned
earlier that there are only finitely many solutions to
max(||a«||,||a2«||)<«-1/2-€
for any €>0, it was not shown until 1979 [31] that there are only finitely many
solutions to
max(||a«||,||a2«||)<iA(«) (9.1.7)
whenever
oo
X if/2(n)<oo. (9.1.8)
n=l
In this formulation, one way of tackling the problem is via overlap estimates which
now give the 'main term' rather than make up the 'variance'. We need to be able
to estimate the measure of
Ui
\r-\j/{n))2 (r + \j/{n))2~\ f(w-^(«) w + ^(«)
m r M n2
*K«))2~| [(m~ ^(n) m + xj/in)
n2 J |_ n ' n
We leave it to the reader to verify the following result.
Let JV> 1, N215 ^A^N. Then the number of solutions to
r2=a(modri), r^n, a^A, n~N
is
« NA.
Remark The value N2'5 can be reduced: we have merely stated a sufficient result
to obtain the next one, which quickly follows.
Theorem 9.4 Let \j/{n) be a monotonic decreasing function of the integer variable
n such that (9.1.8) holds. Then there are only finitely many solutions to (9.1.7) for
almost all a.
9.2 PROOF OF THEOREM 9.2
Let ir(a,«) = ak«fc + ak"1«fc_1H \-anl+n0. Our initial reaction would be to
suppose that the set of a e [a, b] for which
||F(a,/i)||<iAk(/z)
246
Diophantine approximation on manifolds
has measure «\j/\h), and the result then follows from the Borel-Cantelli lemma.
Unfortunately, this does not hold in general, although in a certain sense it does
hold on average. To be more precise, the situation can be retrieved by considering
the intersections of the 'too large' sets (Sprindzuk's method [252] of essential and
inessential domains described below). The obvious example to show what can go
wrong is
F(a,/i) = (wa-r)k. (9.2.1)
Then |F(a,/i)|<i/>k(/z) for
(r-xj/Qi) r+xj/Qi)
ae ,
\ m m
One does not need such an extreme example: problems will arise when F(cai) and
F'(a, ri) are simultaneously small, where F'(tx, ri) denotes the derivative with respect
to a. Since F is a polynomial, this corresponds to two or more roots (which may
be complex) near (or equal) to a. We shall thus be concerned with results on the
spacing between roots of the polynomials under discussion. We first show that we
can assume that F is irreducible over Z (so nothing like (9.2.1) can occur) and
that h=nk.
Step 1: We may assume that F is irreducible
First we need an elementary lemma.
Lemma 9.1 Suppose that P and Q are polynomials of degrees r and s with heights
A and B respectively. Let k = r+s. Then the height of P(x)Q(x) is
>ABk~*\ (9.2.2)
Remark We have not attempted to get an optimal size constant in (9.2.2). It is
clearly trivial that the height of P{x)Q{x) is no more than kAB.
Proof Suppose P{x)=aryf H V a0, Q(x) = bsxs-\ \-b0 with
\ap\=A, \bq\ = B.
Without loss of generality, p+q^(V2)k. Let ct (O^t^k) denote the coefficients
of P(x)Q(x), that is,
I aibr
i+j=t
We establish (9.2.2) by contradication. Assume that (9.2.2) is false, so that
a k
\ct\^ABk~* for all k. We then establish by induction that for each fixed u with
Proof of Theorem 9.2 247
0^w(l/2) ^k we have:
(a) at most one of |<z£6k_£_ J exceeds ABk~4" ";
(b) none of the terms |«£^fc_£_M| exceeds ABk~4""+1.
By our assumption, (a) and (b) hold for w = 0. Now suppose that they hold for all
non-negative integers up to u — 1. Suppose that |tf£6fc_,_J and \ajbk_j_u\ both
exceed ABk~4" ". Then, without loss of generality, i>j. Let t = u+j—i. We then
have
ItfiVi-JHtfA-i-JltfjVj-JltfA-i-J-1
>^ik-2-4">^ik-4",+1-
This contradicts (b) holding up to u — 1. Statement (b) follows immediately from
statement (a) in view of our initial assumption. This proves (a) and (b) by
induction. The lemma follows, since (b) is violated for i =p, u = k — i — q(^k/2). □
Now suppose that Theorem 9.2 has been proved for polynomials of degree up
to k — \. Let $# be the set of a for which there are infinitely many reducible
polynomials of degree k such that
|F(a,ii)|<^(A). (9.2.3)
It follows that, for some /, there are infinitely many polynomials satisfying (9.2.3)
which factorize as F1(a)F2(a), where Fx has degree / and
1/^(001 <^(A). (9.2.4)
a k
Now Fx is a polynomial with integer coefficients and height A ^hk* by Lemma 9.1.
If a is not algebraic, then ^(a) #0, yet the right-hand side of (9.2.4) tends to zero
as h-*co. Hence there are infinitely many different solutions to (9.2.4). Define
pin) by p{n) = \l/(nk~Ak). Then p(A)^\l/(h), since \j/ is decreasing. Also,
£ p(n)*:k*k f>(«),
n^fc4* n=l
and so converges. Hence, by our assumption, |/i(a)| <p\A) holds infinitely often
only on a set with measure zero, and so (9.2.4) holds infinitely often only on a set
with measure zero. Thus X(.srf) = 0, as required to prove Step 1.
Step 2: We may assume that nk=h
We first need another elementary lemma.
Lemma 9.2 Let P(x) be a polynomial of degree k and height h. Then there exists
an integer j with O^j^k and \ P(j)\^ch, where c depends only on k.
Proof This follows from the fact the the coefficients of P are uniquely determined
by the values of P(j), O^j^k. We include a proof for completeness. Since a
248 Diophantine approximation on manifolds
polynomial of degree k can have no more than k roots, the set of k +1 vectors
{(l,...,l)}u{(0,l,2J,...,#): Uy<*}
is linearly independent. Hence the matrix M with these vectors as rows is non-
singular. We note that, if P(x) = F(x,n),
(no,n1,...,nk)M=(P(0\P(l),...,P(k)).
Hence
(«o,«1,...,«k) = (P(0),P(l),...,P(A:))M-1,
and so
max | rij | ^ Cmax | P( j) |,
j j
where C depends only on M, and hence only on k. This completes the proof. □
Corollary For each k^2 there is a constant c such that every polynomial F with
integer coefficients and height h belongs to at least one of the sets
Pj={F:F(j)>ch}, O^j^k.
Now let si <= [a, b] be the set of a for which
\F(oc,n)\^il/k(h)
has infinitely many solutions. Then to show that l{si)=0, we need only show that
/.(sij) = 0 for O^j^k, where
s/j={a e si: | F(a, it) \ < ij/k(h), Fe Pj infinitely often}.
Now F(j+0,n) can be written as G{0,m), where maXj\mj\< Ch for some
C=C{k). Also, m0=F(j, n) > ch. Now let Q(x)=xkG(x~ 1,m). Then Q has leading
coefficient m0>ch. Put Z) = [Cc_1] + l, R(x)=F(xJ) = Q(Dx). Then \lk\ >\lj\ for
0^/^fc-l. Also,
\F(x,l)\<rj(Da)k=>\F(«,n)\<ri,
where <x=j+(Dx)~1. If we let p(h) be a monotonic decreasing function such that
p(h) = Da\l/(h/(DQ),
then
converges. Hence a proof that
\F{0L,ri)\<p\h)
has only infinitely many solutions for almost all a with h=nk (replacingFby —F
if necessary) establishes fi(jrf)=0 as required.
Proof of Theorem 9.2 249
Remark Since we can now assume that h=ak and that F is irreducible, we have
considerable knowledge about the roots of F(a,/i) = 0. To be precise, if we label
these roots as pl9..., pk, then they are all distinct with |pk| ^k. The last inequality
follows from
h\pk\k = \akpk\ = \F(pk,n)-akpk\^khm^x(l,\pk\k-1).
Step 3: The principle of essential and inessential intervals
Suppose that
| F(a, n) | < \j/k{h) for a e J', an interval.
Now consider a larger interval J with X{J) = ^-1(A)A(./')- We might suppose
that \F{a,n)\<K\l/k~l{h) for oleJ. Let tf be the set of all such intervals. We
divide 3tf into two classes as follows:
(i) J is an inessential interval if A(/n/)>(l/2)A(/), for some /e<?f;
(ii) J is an essential interval otherwise.
By Lemma 9.3 below, we need only concern ourselves with those a which
belong to intersections of intervals. In this case, if ae/n/, then \F(a.,n1)\<
Kif/k-\h), \F(ot,n2)\<Kil/k-1(h). Hence R(<x,m)=F(<x,n1)-F(<x,n2) is a
polynomial of degree at most k — l, of height at most h, and with
\R(<x,m)\<K\l/k-Hh). (9.2.5)
We are therefore able to use the case of the theorem for smaller k to show that
there are only finitely many solutions to (9.2.5) for almost all a. Since \j/k~1(h)^0
as h-+ oo, and /?(a,m) = 0 only if a is algebraic, this shows that a can belong to
only finitely many intersections for almost all a.
On the other hand, in case (ii) no point lies in more than three intervals. Hence
the sum of the lengths of the intervals is no more than three times the total length,
say £, of the interval on which it is at all possible for |F(a,/i)| <\j/k(Ji) to be true.
Since by Step 2 no zeros have modulus exceeding k, we have \F(f$,ri)\ ^1 for
\p\>k + l,andsaC^2k + 2. Thus
case(ii) case(ii)
The convergence of X tyih) then completes the proof.
For case (i) the following lemma is vital.
Lemma 9.3 For each positive integer N, let U(N) be a finite set of closed intervals.
Let K(N) denote a subset of U(N) such that
SeK(N)^>3feU(N), ./#/, X(Sn/)^(l/2)A(f).
250 Diophantine approximation on manifolds
Put
s?(N)= (J J, @{N)= (J Jnf.
JeK(N) SeK(N),/eU(N)
./*/,A(./n/)>(l/2)A(y)
Let stf and & be the sets of points belonging to infinitely many s/(N), &(N),
respectively. Then X(0$) = 0 => X{jrf)=0.
Proof We have
oo
m=1N^m
so that X(@) = 0 implies that, for every € >0, there is an M with
X[ (J 08(N)\«l for all Z^M.
We note that \Jz>n>m &(N) is made up of a finite set of closed intervals, and so
can be expressed as a finite disjoint union of closed intervals. Now, if X(Jc\f)>
(l/2)X(J), then J^X, where X is the interval [x0 —t,jc0 + t], where jc0 is the
centre of the interval and t = (3/2)A(/n/). It follows that
X[ (J j/(A0j< 3c for all Z^M.
Hence A(«s/)=0 as required. □
•Ste/? 4: Properties of polynomials
The following result will help us to quantify the statement that a polynomial
is small near and only near its zeros.
Lemma 9.4 Let P(x) be a polynomial with degree k and distinct zeros px, p2,..., pk.
Suppose that 6 is a real number with \px — 6\ ^ \pj—6\ for k^-j'^2. Then
2l-k<iJ*!S , (9-2.6)
l^(Pi)l|0-Pil
<ZH*/
\Pi6)\>22-k\F(Pl)\\e Pl1,. (9.2.7)
IP1-P2
Moreover, if\d — p1\^\p1—pj\ for all j^2, /Ae/i
IP'(P1)II^-P1
<2*_1. (9.2.8)
Proof of Theorem 9.2
251
Proof Let h be the leading coefficient of P{9). Then
k k
P(6)=hfl (6-pj), F(p1)=hY[(p1-Pj),
and so
7X0) <
P(6)
P'iPi)
j=2\Pl Pj
= X\d-p1\, say.
Since
\Pi-pj\^\0-Pi\ + \0-pj\^2\0-pj\,
we have A>21_k to give (9.2.6).
On the other hand, if \6 — px\^\px —pj\, then
lO-pjl^O-Pil + lPi-pjl^lPi-pjl
and so Jr<2*_1, giving (9.2.8).
To prove (9.2.7), note that
W)ip1-p2i=i6i-p1ii6>-p2inp^-
j-3\Pl-Pj
>i2-k\e-Pl\2.
Now, given a polynomial P(x) of degree k with leading coefficient h and distinct
zeros pj,..., pk, we write
P* = |^(p1-p3)---(p1-pk)|-1/2.
Our argument will later split according to the relative sizes of P* and \p1 — p2\-
For this we shall need the following result on the spacing of zeros of such
polynomials.
Lemma 9.5 Let P(x) and Q(x) be two polynomials with degree fc ^ 3, and having
zerospl9...,pk and ol,...,Gk,allof which are distinct and with absolute value at
most K. Suppose that
and
Then
\Pi-pi\^\Pi-Pjl ki-^Kki-tf/l fory>2
IPi-P2I<^*. ki-*2l<G*-
|p1-(71|»min(P*,e*),
where the implied constant depends only on k and K.
(9.2.9)
(9.2.10)
(9.2.11)
252 Diophantine approximation on manifolds
Remark In the case A:=2, the inequalities in (9.2.10) cannot be satisfied, since if
px^p2, we have \p1-p2\^h~1=P*.
Proof Let the heights of P and Q be h and h respectively and put
^ minCP*,0*)'
Our aim is to establish that £»1. We note that the discriminant of P, namely
D(P)=h2k-2Y[(Pi-Pj)2,
satisfies \D(P)\>1. Since
\D(P)\=h-2p*-*\Pl-p2\2 n ipi-pji2.
2<i<j
this (together with the inequality \pr — ps\^2K) gives, for anyy'^3,
\p1-p2\\p2-pj\»P*2h>P*2.
(There would be no advantage in keeping in the factor h.) By (9.2.10) we may
deduce therefore that |p2—Pj\»P* for ally^3. Hence
\Pj-^l\^\Pl-p2\ + \p2-pj\ + \Pl-^l\
^p*+CP*+|p2-pJ|«(l+OIP2-Pjl<2(l+OIPi-Pjl,
using (9.2.10) and (9.2.9). Hence
|/zk-1P((71)|«(l + Ok"2|(p1-^1)(P2-^l)l^"2-
Similarly we obtain
\h'k-1Q(p1)\«a + Ok-2\(p1-<r1)(e2-p1)\Q*-2.
We now use the fact that the resultant of P and Q, that is,
R(P,Q) = \hh'\kYl\Pi-°j\>
satisfies R(P, Q) > 1, and
R(P,Q) = \hk-1P(<71)\\h'k-1Q(p1)\\p1-o1r1 n IPi-^jl
«|/zfc-1P((71)||/z'fc-1e(p1)||p1-(T1|-1|p2-^|.
Hence
l(Pl-^)(Pl-^2)(p2-^)(p2-^2)l»(^*e*)2(l + 04"2k.
Now
IPi-'iKIPi-tfil + IPi-PzK^fl + O,
IPi-^KlPi-^il + ki-^KG'O + O,
Proof of Theorem 9.2 253
and
|P2-^2l^lPi-P2l + IPi-^2l^^*+G*(l + 0^2(l + Omax(P*,e*).
Hence
Cmin(P*,g*)(l + 03/>*S*max(P*,g*)»(l + C)4"2k(P*e*)2,
and so
c»a+o4-2k(i+o-3,
which gives C»l as desired. □
Proof of Theorem 9.2 We assume that the result has been established for £: —1.
The case k = \ is a simple application of Lemma 1.2. We note that, since \]/(h) is
monotonically decreasing and
Z MO
h=\
converges, we have
(2A + l)^(2A + l)<(2A + 2)^(2A)<2(^(A)+--+^(2A))->0 as/z-»oo.
Thus h\l/(h)-+0 as h-+oo.
For any F(a,/i) which is irreducible and with «fc=max|«7| =/z, write pl,...,pk
for its zeros. We need only consider the set $f of 6 with 16—p x | ^ 16 — pj | for j ^ 2,
since if there are infinitely many solutions to (9.1.5) on a set ^" with positive
measure, then there is some ordering of the zeros of each of the polynomials F(a, ri)
such that ¥ r\3~ has positive measure. We also recall that we can restrict 6 to the
interval [—k — \,k + l].
Now write
t = min | p 1 — pj | = | p 1 — p21 (without loss of generality),
v = 2k|F(Pl,/i)|-Vfc(/z)
(so that IFXfl.itfl^MHfl-Pi^v by (9.2.5)),
^ = min(v,(tv)1/2).
The idea will be to show that
X({oc: aeJQi) for infinitely many h}) = 0,
where
J(h) = {a 6 R: | a - px | < fi, px e @(h)},
«(/r) = {p:F(p,ii) = 0,F(a,Ji)e^(A)},
^(/z) = {F(a,«)eZ[a]: F is irreducible with h=nk^\nj\ fory <k].
254 Diophantine approximation on manifolds
We split ^(h) into two classes:
Fe^^ot^?*, Fe@(h)oT<P*.
It then suffices to prove that 2({a: ole/Qi) for infinitely many /z}) = 0, and
2({a: oleXQi) for infinitely many h}) = 0, where
/(/z) = {a6R:|a-p1|<//,F(p1) = 0,F6^(/z)},
with X(h) defined similarly with <%(h) replacing jrfQi). By the remarks after
Lemma 9.5 we have @{h) = 0 if k = 2, and so X(X~(h)) = 0 automatically in this
case.
Given a polynomial Pes#(h), we write
I'(P) = {oleU: \oc-p1\<fi}
and
I(P) = {oleU: \oc-p1\<ml/-\h)}.
We note that I'(P)<=I(P) and X{r{P)) = \j/{h)X{I{P)). We then define the sets of
inessential and essential intervals respectively by
#={/(/>): X(I(P)r\I{Q))>\X(I{P)) for some Qes?{H)},
$= u /(P)\#.
Fej/(/i)
77*e inessential intervals Suppose that I(P) e # with 2(/(0 n/(P)) > (1/2)X(I(P)).
On /(P) we have
by the definition of s/(h). We can thus apply Lemma 9.4 and obtain
\P(<x)\^2k-1\P'(p1)\\J/~\h)v=22k-1il/(h)k-1.
Since analogous results hold on I(Q), we obtain, for oceI(P) n/(0,
\R(a)\<22kiJ/(h)k-\
where R(x) is the polynomial P(x) — Q(x) of degree not exceeding k — 1, and height
at most h. By our inductive assumption, there are only finitely many such R for
almost all a. Hence almost all a belong to only finitely many intersections
I(P)nI(Q), and so, by Lemma 9.3, almost all a belong to only finitely many
I(P) e S.
Proof of Theorem 9.2
255
The essential intervals Let
X\K)= |J I\P\
Hence
W(h))^(h) £ A(/(/>))
/(F) e <?
^3\j/(h)M U /(/>))^3iA(A)(2A: + 2).
\/(P)e<y /
Since
h=l
converges, almost all a belong to only finitely many cfih), and so to only finitely
many I\P) when I(P)e£.
Completion of the proof We now consider Pe&(h). We first note that
ix2^xv = \P\p)\-^\h)2k\p1-p2\
^/z1-k|P'(p)l"12(1/2)k|p1-p2|iA(/z)
= 2kP*2iJ/(h)^2kP*2. (9.2.12)
Let
#(/)= U *0).
4-,'_1</i<4J'
<#(j,l) = {Pe<#(j): 2-'<P*^2-'-1}.
Lemma 9.6 7%e number of polynomials Pe^(jJ) with I'(P) ^0is « 2/ +1.
Proof If I'(P) ¥" 0, then p must have imaginary part less than ix, by the definition
of I'(P). If there were C(2l +1) such P, then there would be C(2l+1) roots p with
imaginary part less than \i, and hence (by the pigeon-hole principle) two roots a
distance
<5<(^2 + 4/c2(C(2'+l))2)1/2
apart. Thus, by (9.2.12),
5«C2~l.
However, by Lemma 9.5 (recall that t<P* in @{h)\ roots are »2~l apart. Hence
C«l, which establishes the lemma. □
256 Diophantine approximation on manifolds
We now note that
F€<if(j) I PeV(j,l)
«X(2' + D//^Z(2/ + 1)(vt)1/2
«X(2/ + l)(4-/ + 1iA(4j"1))1/2 (by (9.2.12))
Since
00
1*2
-k
k=l
converges, it follows from the Borel-Cantelli lemma that almost all a belong to
only finitely many I(P), when PeMQi). This completes the proof of the inductive
step, and the result follows by induction. □
9.3 PROOF OF THEOREM 9.3
First we note that we can restrict s to an interval on which y(s) is monotonic. We
can then put <x=y(s), jc(j)=/(a), where ae [A,B]. We have
which is therefore non-zero for almost all ole[A,B]. Indeed, neglecting a set of
measure € we can assume that |/"(a)l^c( = c(€)), for a belonging to a finite
number of subintervals of [A, B] (we required the existence and continuity of
x(3)(s) and yl3)(s) here). Thus we can henceforth suppose that /"(a)^c for
ae[A,B]. We may also suppose that A>0. We consider
l"i/(a)+«2a + "3l<^~2~€> Kmax|/ij|</r. (9.3.1)
Write F{nf ot)=n1f(ot) + n2tx+n3. Let
M=l+ max 1/(001.
ze[A,B]
It suffices to consider separately the two cases:
(i) \ni\<\n2\K2M); (li) \ni\>\n2\l{2M).
In case (i) we have
\F(n,a)\ = \nina)+n2\>i\n2\. (9.3.2)
Since /"(a) ^c>0, there are at most two intervals on which \F(n,tx)\<h~2~e. Let
Proof of Theorem 9.3 257
j5 be the midpoint of one of them, say J. By the first mean-value theorem,
F(n,y)=F(nJ) + (y-p)F'(n,0,
where /?<(<y, or y<C<p. Hence
If we let C=max|/(a)|, we must have |«3|^(C+B)\n2\ from (9.3.1). Hence the
sum of the measures of intervals for which (9.3.1) can hold in this case is
oo
-3-€
\n2\
n,= -oo |n,|sS|n2|/(2M)
*i*0 |/!3|<(C + B)|n2|
which converges. Thus, by Lemma 1.2, there are infinitely many solutions to (9.3.1)
only on a set with measure zero in case (i).
We can now consider (9.3.1) replaced by
\F(n,x)\<n;2-\ |«2|,|/i3|«fi1, U«x. (9.3.3)
The rest of the proof will rely on the second mean-value theorem:
F(/i,a)=F(/i,j9) + F(/i,i9)(a-i9) + F'(/i,0^^-, (9.3.4)
where, since F"(/i, £) = «i/"(£)> we can assume nl«F"«nl. We shall consider
\F(n,oc)\<n;v, U«!, l"2U"3l«"i> (9-3.5)
showing firstly that there are infinitely many solutions only on a set with measure
zero if v = 6. We shall then show that if (9.3.5) holds infinitely often on a set with
meaure zero for v = u, then the same conclusion holds with v replaced by
2 + 2(w — 2)/3. This will complete the proof by induction.
Step 1: The case v = 6
First we require a simple lemma which will be used several times in the
following.
Lemma 9.7 Let g be defined on [C,D], with a positive continuous second
derivative, and write d x = min | g'(x) \,S2 = min | g "(x) \. Then, for every P>0,we have
UP): = K{ae[C,D]:\g{a)\<P})
<<minK'(S1/2)- (916)
Corollary Let J be a subinterval of[A,B], 5=minag/ |jP(ji, a)|.
258 Diophantine approximation on manifolds
Then
X({txeS: \F(n,oc)\<p})<<mm&(-P) V (9.3.7)
Proof We obtain L(/?)« /?<5j~* simply from the first mean-value theorem as in an
argument above. Now suppose that (fid^ 1)1/2<j9(5j"1. Split L(/?) into at most three
intervals on each of which either rmn\g'(oi.)\<3\l2p112 or min|^'(a)| ^bx2l2$vl-
Any intervals with mm\g'(oL)\^S\12 P112 have length at most (/?/<52)1/2 by the first
mean-value theorem. On the other hand, an application of the first mean-value
theorem to g'(a) on an interval / with \g'{oL)\<Pmd\!2 gives A(/)<(0/<52)1/2,
which completes the proof. The corollary follows immediately, since
nl«F"(n,(x)«nl. □
Now the set of a for which \F(n, a)| <«j~6 has measure, by the above corollary,
at most
' -6\l/2
—) =*r7/2.
An upper bound for the sum of these measures over all n is thus
f *r7'2 I i« f «r3'2.
which converges. The proof is then completed by a standard appeal to Lemma 1.2.
Step 2: The inductive argument
It should be clear from Section 9.2 and the above that problems arise when
|F| and \F'\ are simultaneously small. It suffices to show that (9.3.5) has only
finitely many solutions for almost all a in each of the following four cases, where
the division depends on the size of \F'(n, a)|. We assume that the result has been
proved with v=u, and now let v = 2 + (2/3)(u—2). We also let n=nv to simplify
notation.
Case (i): nll2^\F'(n,a)| (we automatically have \F'(n,a)| = |«/'(a)+«2|««,
of course). Let us fix n,n2 and consider the solutions as n3 varies. For each n there
are at most four intervals on which \F(n, a)| ^n~v, \F'(n, a)| ^ n1'2. Let /? be a value
of a in such an interval with \F'(n, a)| minimal. Say /? belongs to the interval
jf = J(n3). Then X(J)«n-v\F(nJ)\~1=n~vT, say. Let
£2 = 1+ max |/"(a)|.
ae[A,B]
Write
f = f(n3) = Le[A,B]:\oc-p\< Z
4K
Proof of Theorem 9.3 259
and so X(J)«n~v2.(f), J c /. On / we have, by (9.3.4),
|F(ii, a^^lFCii.^l+^T-^f |i^(ii, a)|r^T<^.
If n' = (n1,«2,«3+w), this gives \F(n',cc)\ = \F(n,cc) + m\>m — \/2.
Hence / (n3) n / (n3 + m) = 0. Thus
l([)S(n3)\^B-A9
and so
^2(/(«3))««-p.
Since
I *~P
n,n2
n2«n
converges, this settles the first case by Lemma 1.2.
Case (ii): 1/2 ^ |F'(n, a)| <«1/2. We now fix just nl9 and allow n2 and n3 to vary.
We make use of the essential/inessential interval technique of Section 9.2. Note
that in case (i) all intervals were essential. Define /? and t as in case (i), and put
f = {oce[A,B]:\oi-p\<(nT)-1}.
For ae/ we have, by (9.3.4),
nz (rn)
We call an interval # essential if there exists another interval /"' with 2(/n/')^
(1/2)A(^), and inessential otherwise. Now suppose that two intervals #,#'
intersect for integer vectors n,n' = (n1,n2 + b,n3 + c). Then, for ae/n/' we have
|F(/i,a)-F(/i',a)|««-1,
that is,
|^a-c|«rt_1. (9.3.8)
However, since we have |F'(n,a)| <«1/2 in this case, we have
|F'(/i,a)-F(/i',a)|<2rt1/2,
and so \b\<2n112. Hence (9.3.8) has infinitely many solutions only on a set with
measure zero, by Lemma 1.2. The set of a belonging to infinitely many inessential
intervals therefore has measure zero, by Lemma 9.3.
260 Diophantine approximation on manifolds
To deal with the essential intervals, we note that
2(^)««-t'T-1=^2(/),
and # cz J. Since the measure of the union of the essential intervals is «1, and
Ynl~v converges, this completes the proof for this case with the usual appeal to
Lemma 1.2.
Case (iii): n1~v/2< \F'(n, a)| <l/2. We argue as in case (ii), except that we put
S = {aelA,B]:\a-p\<8}t
where
^_w-max(l.(r-l)/2)
and there are now no inessential intervals, since the inequalities
|F(n,a)|<i and |F(/i,a)-F(/i',a)|<l
give b = c=0.
Case (iv): \F'(n, a)| ^n1 ~v/2. We now suppose k2 ^n<(k + \)2 and consider the
set of intervals arising from n with \F'(n, a)| ^/i1-t;/2. Our intervals to divide into
inessential/essential classes now have the form
f = {oce[A,B]:\oc-p\<k-%
where w = l + (l/2)w (note that v>w). For ae/ we have
\FXn,<x)\^\F'(nJ)\ + Kk2\a-P\«k2-w. (9.3.9)
Hence
\F(n,tx)\«k-2v+k2-2w«k2-2w. (9.3.10)
At this point we appeal to the truth of the result for u. We note that if / and f
correspond to (nlt n2, n3) and (nl+a,n2 + b, n3 + c) respectively, then (9.3.10) gives
\af(<x) + b<x + c\«k2~2w,
while k2^n<(k + \)2 and (9.3.9) give
\a\, \b\«k.
Hence the measure of the set belonging to infinitely many such intersections has
measure zero. (Here w^l + (l/2)w was vital).
Turning our attention to the essential intervals, we obtain
/lO/K^""'-1)1'2 (by Lemma 9.7)
«k-v~1 =k~1+w-vHf).
Hence the set of intervals J for a fixed k has measure «k~1+w~v. Summing over
k gives a convergent sum and so completes the proof by Lemma 1.2. □
Notes
261
We finish this section by pausing to reflect on the above proof. The four cases
at the end can be made to overlap in application. Indeed, case (iii) could be
replaced with an alternative argument (see [35]). There is no problem in replacing
n~v with «-1i^(«), where
n=l
converges, except in case (iv). To deal with this case when proving their best
possible result, Bernik et al. [35] needed to refer to other work of Beresnevich and
Bernik [28], where it was shown that the simultaneous inequalities
|F(w,a)|<«-2, |F(«,a)|<«-1/4
have only finitely many solutions on a set with measure zero.
NOTES
Related results are discussed in Chapter 2 of [254]. For example, one can consider
systems of forms in place of our single form. V.I. Bernik and M.M. Dodson are
currently writing a monograph on this topic. Many results have been proved on
the dimension of the exceptional sets for these types of problems. For example,
see [13], [14], [15], [33], [48], [49], [73], [74], [235]. We have not dealt here with
the variation (compare Theorem 4.6)
n1+*Y[ \\™J\\<1
which was conjectured to have only finitely many soultions for almost all a by A.
Baker (see page 96 of [4]). This problem was solved by Bernik [34]. One
particularly important new development is the work of Kleinbock and Margulis
[160], who use ideas from flows on homogeneous spaces.
10
Hausdorff dimension of exceptional sets
Non-normal numbers. Exceptional sets in uniform distribution. The
Besicovitch-Jarnik theorem. Generalizations with applications to the
Duffin-Schaeffer problem and a two-variable problem. An exceptional
set from Chapter 8.
10.1 INTRODUCTION AND STATEMENT OF RESULTS
Until now we have concerned ourselves only with what is true for almost all
numbers. We have not investigated the size of the exceptional sets, except to give
one or two examples. The natural measure of size of an exceptional set is its
Hausdorff dimension. For a full introduction to the general area of Hausdorff
measures, see [230]. We shall be concerned here only with Hausdorff measures
using the class of functions Xs (O^s^l), and for this purpose the following
definition suffices.
A set Sf of real numbers has Hausdorff dimension d (written as dim Sf=d)
if the following two conditions hold:
I. For any p>d, €>0, there is a sequence of intervals ,/• such that
oo oo
&c[JSj> £ KSy<l,Kfj)<e for ally. (10.1.1)
II. For any /? < d there exists € > 0 such that no sequence of intervals Ji can satisfy
all three statements of (10.1.1).
Some authors would call the number d defined above simply the dimension
rather than the Hausdorff dimension [81]. It is not hard to verify that all real sets
have such a dimension, and the value obtained must be between zero and one.
Also, for any sequence of sets &-p we have
dim( (J ^.U sup dim f,-. (10.1.2)
In particular, if J^is a countable set, then dim(Jr) = 0, and dim((f u J^) = dim^
for any real set S.
Introduction and statement of results 263
It is clear from the above definition how our investigation should proceed. We
shall be looking at coverings of the exceptional sets and investigating whether
£ KSjXh (10-1.3)
or, starting with (10.1.3), we shall deduce that the J} cannot cover the set under
discussion. These basic ideas will be clearly seen in the proof of our first result on
digits in the representation of numbers to base r.
Theorem 10.1 Let r^2, and let s£ be a non-empty subset of {0,1,..., r—1} with
s=\s/\. Write @t for the set of numbers which only have digits from stf in their
'decimal' expansion to base r. Then
logr
Corollary The set of numbers which are not entirely normal to base r has dimension 1.
The corollary follows, since if a is entirely normal to base r, it is simply normal
to each of the bases r", and therefore there are sets of dimension
logO"1-!) ,
——— >\ as«->oo,
log(r")
which are not entirely normal to base r.
A modification of the proof of Theorem 10.1 yields the following stronger
result.
Theorem 10.2 Let r^2. Then the set of numbers not simply normal to base r has
dimension 1.
Of course, it follows a fortiori from the above that the set of non-normal
numbers has dimension 1. We remark that Eggleston [80] proved that given non-
negative reals p0i...9pr_1 whose sum is one, if we use A(j, r, N) as in Chapter 1 for
the number of occurrences of the digit j in the first N places of the expansion of a
real a to base r, then the set
{<x:A(j,r,N) ^/^JVas N -> oo for O^j^r—1}
has dimension /? given by
Thus P = 1 if and only if each pj = r~1.
From Theorem 10.1 or Theorem 10.2 it follows that the set of a for which {<xrn}
is not uniformly distributed modulo one has dimension one. The following result
of Erdos and Taylor [95] generalizes this to any lacunary sequence.
264 Hausdorff dimension of exceptional sets
Theorem 10.3 // an is a lacunary sequence of integers, then {ccan} is not uniformly
distributed modulo one on a set with dimension 1.
They also proved the following theorem as a complement.
Theorem 10.4 Let an be an increasing sequence of integers with an=0(nc)for some
c^l. Then {ancc} is not uniformly distributed modulo one on a set with dimension at
most 1—c_1.
Remark We have already seen that {no} is uniformly distributed modulo one for
all irrational a. The corresponding result for {pa.}, where p runs throught the
primes, follows from work of Vinogradov (see page 180 of [271]). It follows from
Theorem 10.4 that for any sequence an with an=0(n\ogri), the exceptional set still
only has dimension zero. There are sequences with an = 0(n) but for which the
exceptional set of a is uncountable (Theorem 11 of [95]). On the other hand, if
an=nk, the exceptional set consists just of the rationals [280].
We now consider the sets of measure zero for which
has infinitely many solutions when
X>(")
n=l
converges. Jarnik [146] and Besicovitch [38] proved the following result.
Theorem 10.5 Let £f be the set of real numbers for which there are infinitely many
solutions to the inequality
||wx|| <n~',
where /?>1. Then
dim y=-^-:. (10.1.4)
P + l
Remark It follows from this result that the set of 6/y satisfying (7.1.11) has
dimension zero. It is only a little harder to prove the following result instead of
Theorem 10.5, which then follows as a corollary.
Theorem 10.6 Let s£ be an infinite subset of the positive integers, and write
y = sup<0^/z: £ n~h diverges>.
I ne.a/ )
Let p>y. Then the set Sf of a for which
\q*-p\<q-\ (p,q) = h qes/ (10.1.5)
Introduction and statement of results 265
has infinitely many solutions satisfies
dim^=j±Z. (10.1.6)
1 + p
Moreover, the result remains valid with the condition (p,q) = \ removed from
(10.1.5).
Remark Theorem 10.5 was proved in a more general form (with s simultaneous
inequalities in place of (10.1.5)) by Borosh and Fraenkel [47]. Rynne [235] further
generalized the result to consider, for a set of vectors ^<=Z\ the simultaneous
inequalities
k
Z 4j«ij
<\q\~\ qe@, X^i^m
for t > y, where
y = sup<h: £ \q\ h diverges >.
From Theorem 10.6 we may immediately deduce the following.
Corollary Let s4 be any infinite set of positive integers, and define Sf as in
Theorem 10.6. Then dim^^l + p)-1. In particular, the Hausdorff dimension of
the set of a for which there are infinitely many denominators in the continued-
fraction expansion which belong to s£ is at least \.
We state another corollary as a theorem in its own right.
Theorem 10.7 Let ^(«) be a non-negative function such that
oo
Z#(»)
11 = 1
diverges. Then the set of oc for which
\om-m\<\j/{n\ (m,n) = l (10.1.7)
has infinitely many solutions has Hausdorff dimension 1.
Remark This result demonstrates that the Duffin and Schaeffer conjecture
cannot 'fail badly'. The reader should note that we do not require the divergence of
11 = 1 n
It follows that the set of exceptional a for Theorem 2.8 has Hausdorff dimension 1.
As a final result of this type, we shall explain how to modify Theorem 10.6 to
obtain the following.
266 Hausdorff dimension of exceptional sets
Theorem 10.8 Let p ^ 1. Then the dimension of the set of cc for which
\<xp—q\<p~p, p,q both primes,
has infinitely many solutions is 2(1 + p) ~ \ that is, the same dimension as with p and
q unrestricted.
Combining this with Theorem 6.1 produces the following corollary.
Corollary The set of ol having infinitely many convergents to their continued-
fraction expansion with numerator and denominator simultaneously prime has
dimension one, but Lebesgue measure zero.
Finally we prove the following result on one of the exceptional sets of Chapter 8.
Theorem 10.9 If an+l—an^c>0 for all n, then the set ¥ ofa for which [a0-] is
only finitely often prime has Hausdorff dimension one.
10.2 PROOF OF THEOREMS 10.1 AND 10.2
By (10.1.2) it suffices to consider the set <T = ^n[0,l). We first prove (10.1.1) for
/?> log si log r. Write
n yLr"' r" J'
A =
where the union is taken over all integers a with O^a^r" — 1 such that a is
composed only of digits from stf. Then each £n is composed of sn closed intervals
of length r~n, and </„<=</„_!. We write
/= n a-
n=l
Then # is non-empty since it is the intersection of a nested sequence of non-empty
compact sets. Also, </ = #u«^', where #ls empty if r—l$<s/, and otherwise it is
the countable set of rationals which have a finite 'decimal' expansion in base r
with last digit b satisfying b — \ejtf, b$jtf. Clearly each £n is a finite covering of
f. Given €>0, choose n so that r~n<€. If we label the intervals making up tfH
as Sp we then have A(./j)<€ and
YdUSj>=snr-*i<\.
j
This establishes (10.1.1) for /?> logs/log r.
Now suppose [l< logs/log r, that is rp<s, so that there exists N' such that
(r+l)rpN
sN>K _/ f for N>N'. (10.2.1)
sr p — \
Proof of Theorems 10.3 and 10.4 267
Choose N' accordingly, and, in order to establish condition II, we now pick
e = r~N\ so that (10.2.1) holds when r~N<£. We note that if Ji is a sequence of
intervals with
then the number of intervals with l{J^>r~n is no more than rn/?. We construct
closed sets Jf„, where X n = Jn when r~n ^ €. For larger n we vary the
construction, to ensure that X nJ = 0, X # 0, where
X= f]Xn, J= \]Jy
n=l j=l
It follows that X is not contained in J, and so the Ji do not cover #. This will
complete the proof once the construction is completed.
We choose Xn<= Xn_ x so that no interval of Xn has a non-empty intersection
with those intervals J^ with
r~n^X{J])<r1-n.
The number of intervals in Xn which each Ji intersects is at most r + 1. If we
denote by Kn the number of intervals in Xn, this gives
K^sK^.-ir+l)^. (10.2.2)
Since Kn=sn for n^N, and in view of (10.2.1) and (10.2.2), this gives
for all n^N, by induction. Thus A^„>0 for all n, and so X^0, since it is the
intersection of a nested sequence of non-empty closed sets. Since ae/=>a^X'„
for some «, we have X nj= 0, and the proof is complete. □
The above proof is easily modified to establish Theorem 10.2: we need have
only one digit missing in the N to N{\ + rj) places for infinitely many N, where
?7>0 is fixed, to show that a number is not simply normal to base r.
10.3 PROOF OF THEOREMS 10.3 AND 10.4
To prove Theorem 10.3 we need only show that (10.1.1) cannot hold with /?<1.
The idea is to find a set £ of a on which there are too many solutions to the
inequality {antx} ^ r\ for some rj. We choose rj and another parameter v = 17(17) as
follows. The reason for the choice will become apparent later. First, since an is a
lacunary sequence, there is a number p>\ such that
^±i^p for all n.
268
Now put // = (1 — f$) ~l, and write
Hausdorff dimension of exceptional sets
v =
fi\og(4/rj)
logp
» bn = anv
(10.3.1)
(the idea is to get bn+1bn * sufficiently large), and pick 77 < 1/10 so that
2rjv<\,
Pick € = b2 l . Let S0 = [0,1),
'. = '«-inU
m m+
-u.*. *
and put
00
$= n *..
n=l
Then ^ is a non-empty set, and <xe£=>{bnot.} ^n. Hence
N N
£ \^-+0(\)>2nN+0(\).
n=l
{<**<*} <£l
V
Thus {antx} is not uniformly distributed for cueS.
Now suppose J j is any sequence of intervals with
00
£ A(.//<1, /l(^)<€.
We now define subsets J* of #„ such that #"# 0, !Fr\J> = 0, where
00
00
n=l j=l
Let ^0 = S0,^\=SV Pick, for w>2,
where * indicates that the union is only over those m for which
m m+n
Pn bn
njj=0
when
b;l<X{J3)^b;}u
(10.3.2)
(10.3.3)
(10.3.4)
and each interval in J*j lies completely within an interval of SFn_ 1# Let the number
of intervals of length rjb ~1 in J^ be Nn. To complete the proof we need only show
Proof of Theorems 10.3 and 10.4
that Nn remains positive. We have
where
269
(10.3.5)
gn-
*-r
and Rn is the number of rejected intervals which overlap with those J} satisfying
(10.3.4). Let 01 be the set of j for which Ji overlaps with a rejected interval. Then
by (10.3.3) and (10.3.4),
Substituting this into (10.3.5) then gives
Nn>(r,gn-3)Nn.1-2gM-i
Now, from (10.3.1),
so that, if
3 3^
S„ 4
then
^.-i>:^i.
f
^.>a--«-i- J*f ftwi"') > J*f.
using (10.3.1) again. Since Nt =bx ^4/r]b{, this establishes that
Nn>-b*n
V
for all n, by induction. This completes the proof of Theorem 10.3.
□
In view of (10.1.2) and Theorem 5.6 (the Weyl criterion), in order to prove
Theorem 10.4 we need only show, for each A > 1, that the set $ of a 6 [0,1) for
which
1 N
- £ e(M
iV n=l
+-►0 asiV-»oo
(10.3.6)
270
Hausdorff dimension of exceptional sets
has dimension at most \—c 1. That is, we need only demonstrate condition I for
& withj5>l-c_1. Write
s=
1
€=6?
p-{c-l)(l-P)'
For M>\ (not necessarily an integer), write
S(Af,a)=^ Z e(aAfl„),
and let J5" be the subset of [0,1) for which
|S(M,a)|>2Ar€
for infinitely many positive integers M. Then, clearly (f <=#", and if NS^M<
Ns+1, where N is sufficiently large, then
\S(N*,a)\ = N-*\MS(M,a) + d{M-N*)\
(where |0|<1)
Hence &^<g{K\ where ^(#) is the set of ae [0,1) for which
\S(N\ a)| ^ ATes for at least one 7V>#.
We now consider a covering of the set of a e [0,1) for which
|5(Af,a)|>i/.
We have
(10.3.7)
(10.3.8)
da
^Z)MC,
where 2nhan^Dnc. Let H=DMC. Suppose that (10.3.8) holds, then
\S(M,tx + d)\ >jt] when |<5|<?//2//. We consider a covering of [0,1) by intervals
of the form
[2//' 2# J'
7 = 0,1,2,
Let Jlt J2,..., J be those intervals (ordered in increasing size of left-hand end
point) on which (10.3.8) holds for at least one a, with J having 1 as its right-hand
end point if necessary. We then have
V
2 q
|S(M,a)|2da=—.
o
M"
Proof of Theorems 10.5, 10.6, and 10.7 271
Thus
ir^-'O^D<Tj andso q^ZHrj-'M-' + i.
4 2H M
By Holder's inequality,
q / q \1-P/ q \P
^16^-3M"1iy1-^.
We can thus cover the set for which (10.3.7) holds for a single N with intervals
Jj with
X USjY< 16N~ 1/2N~Wc(1 "*)s< 16AT-3'2
i=i
by our choice of s and €. Hence, on choosing K sufficiently large, we can cover
the set for which (10.3.7) holds with each k{J^ as small as desired, and with
I A(.//<1.
This completes the proof. □
10.4 PROOF OF THEOREMS 10.5,10.6, AND 10.7
Let 9" denote the set of a for which (10.1.5) has infinitely many solutions with
the condition (p, q) = 1 removed. Then Sf c $P. It then suffices to prove conditions
I for $P and II for Sf. Let /? > (1 + y)/(i + p). Give € > 0 we now exhibit a covering
of ST satisfying I. Pick X so large that (2Xyip+1)p<£ and
X 2V(P/I+/,"1)<1- (10A1)
qe.af
q>X
This is possible since pp + p — \>y. We can cover Sf'n[0,l) with
p-q~p p + q-p~
u ['
oo
= (J ./,, say,
OsgpsSg-1
and
00
£ ^(j^/= £ 2V°"+'-1)<i.
This gives I as required.
j=l q>X
qe.a/
272 Hausdorff dimension of exceptional sets
Now we let /?< (l + y)/(l + p) and show that I cannot be satisfied for all
sufficiently small €. We first need two simple lemmas.
Lemma 10.1 Suppose that J is an interval of length £ and let 0<<5<^£. Let
N(q, J) denote the number of intervals
J(p,q,S) =
,q q
(P,q) = l
which intersect J. Then
(i) N(q,J)^2£q+l; (10.4.2)
(ii) if £>q"-1 for some rj>0, and q>qQ(rj)i then
N^^^C^Ccpiq) (10.4.3)
and the number of intervals lying wholly within J is
>C2(rj)£<p(q). (10.4.4)
Proof Clearly N{q, J) is no more than the number of fractions plq in an interval
of length C+2<5, which is no more than q(£+2S) +1 <2£q +1, to give (10.4.2). The
bounds (10.4.3) and (10.4.4) follow simply from the estimate
a^p^a+b q
(p,«)=l
forany?/>0. □
Lemma 10.2 Let Q^\, and let & be a set of positive integers not exceeding Q.
For (5 > 0 write
<0 = {S(p,q,d):qe<%,{p,q) = hl^P<q-l}.
Then
U4(£|^|)2<5. (10.4.5)
0
Proof The left-hand side of (10.4.5) is
Z Z ui I i.
q,se& l<p^-l q,se& p,b
l^r^s—1 ps = b(modq)
0<\(p/q)-(r/s)\^2d l^\b\^2SQ2
(p,q) = (r,s)=l
Now, if d = (q, s), then d must also divide b in the congruence ps=b(modd). Hence
the number of solutions to the congruence is
2d[2SQ2d-1]^4Q2S.
Summing over q and s then gives (10.4.5) as desired. □
Proof of Theorems 10.5, 10.6, and 10.7 273
Proof of Theorem 10.5 Let
00
where the Ji satisfy the second and third statements of (10.1.1). Pick g satisfying
O^g^y with (l+g)/(l+ /?)>/? and such that
ne.n/
~g
diverges. Hence we can find infinitely many K such that
K8
I l>i—27?- <10A6)
(l/2)K<n^K
nes/
log2*
We let ^0 = 1 and pick K1 as a value satisfying
<piq)
log2 2q
>2q1-" for q>Kx,
(2CAr\) + $\lh
K^Q^zn&K^i £2 \ , (10.4.7)
where rj is picked for Lemmas 10.1 and 10.2 as 1/4((1 + g)/(\ + p) — /?)• Put € = K^ *.
Fory'^2 we pick Kj>max(2Kj_ uTj), where
1 \ cm ) '
and also satisfying (10.4.6). We now construct our nested closed sets
[0,1] = e/o=)</i3 • • •sucn tnat> f°r everyy >1, the following four conditions hold:
(i) ^. is the disjoint union of M-} closed intervals of length ^-j — KJ1 ~p;
(ii) fj meets no interval of ^with length between £j and €j_ t;
(iii) if a e /j5 then there is a. qes/ with ^A^- <q^Kj_1 and
la-/*/^*;1-*, (/>,$)=i;
(iv) Mj>K)+g-2tl.
Hence, if
/- n /j.
we have c/# 0, fg:^,/^^. This will complete the proof once the inductive
step has been verified.
274 Hausdorff dimension of exceptional sets
We suppose that (i)-(iv) are true fory and proceed to construct fj+ x satisfying
the conditions. It should be clear that, by our choice of Kl9 they hold for/ = l,
and thus our construction is valid by induction. By our choice of Kj+ x we can
find a subset ^<=j/ with
qe@=>iKj+1<q^Kj+1
and
Kg
q% log2 Kj+l
We now construct </,+ i from intervals
l-lill p | €J+1
q 2 ' q 2
This ensures that (i) and (iii) are satisfied for/+1 once we remove any overlapping
intervals. Since these intervals must lie entirely within ^-, the number of possible
intervals is, by (10.4.4),
>C2(n)MJej'Z<p(q). (10.4.9)
Here we noted that £j >IC]+1 by our choice of Ky To ensure that we have disjoint
intervals, we need only remove at most
2K);{\a\2 (10.4.10)
intervals, by (10.4.5).
Now let
To obtain ^rjn^j+1 = 0 we need only remove, at most, a further
X Qoyu (jo !>(?)+ I £(2;i(jo<7+i) (10.4.11)
intervals from (10.4.2) and (10.4.3). Here J21'}1* denotes the subset of intervals in
J*j with length exceeding ICj+l and JrJ2) the remaining intervals. Now, by (10.1.1),
the first sum in (10.4.11) is
^CMXT'-4*!. Ml)- 00.4.12)
Proof of Theorems 10.5, 10.6, and 10.7 275
Likewise, the second sum in (10.4.11) is
*Z3Kj-+3! Y, <p(q), (10A13)
qe&
since (p{q)>2ql~nlog22q for q^K1.
Combining (10.4.9,10,12,13), the number Mj+ x of intervals available is at least
We note that
2Kj-f\m2 ^ 2K);r2g ^ 2K}:r*n ^ kj-*? x <m>-
qe#
Hence
MJ+1 >\C2{n)K9j-p-2n>K}+9-2\
using the definition of Kj+1 (note in particular that Kj+1^Tj+1). This completes
the inductive step and so completes the proof. For the case j = 1 there are no
intervals of & that intersect fXi so the first step is simpler. □
Proof of Theorem 10.7 We follow the above proof with y = 1. In place of (p, q) = 1
we have p as a prime, and so in place of <p(q) we have n(q) ^ qlXogq. The bound
corresponding to (10.4.3) is a simple consequence of Lemma 6.4. The bound
corresponding to (10.4.4) requires a result such as [142] on primes in short
intervals, and this forces the r\ here to be >7/12 (or > 0.535 if one uses the best
known lower bounds for primes in short intervals). This forces us to pick the Kj
spaced further apart, but does not introduce any serious difficulty into the proof.
Proof of Theorem 10.6 Let € > 0 and put
j3f(a) = {«:«-(a+1)£^»A(rt)<«-a£}forUa^[€-1] + l.
Also, write
j*(0) = {n: ^(/i)</Te}.
For at least one a,
276 Hausdorff dimension of exceptional sets
diverges. We can thus find a subset @<^s#{a) such that
neJi
diverges for h^at, and converges for h>ae. We can then apply Theorem 10.5
with p=a€ + €. Then there are infinitely many solutions to
\ccn—m\<n~p with (m,ri) = \
for a belonging to a set with dimension
l + ae
l+ae + e
>l-€.
Since n p=n fl€ €^^(«), this gives infinitely many solutions to (10.1.7) on a set
with dimension at least 1 —2€. Since € was arbitrary, this completes the proof. □
10.5 PROOF OF THEOREM 10.8
To prove Theorem 10.8 we need only show that, for j5<l, there is an €>0 such
that condition I fails. We suppose that a sequence of intervals is given which
satisfies the second and third statements of (10.1.1). For the sake of clarity we shall
suppose that an = n; the more general case follows by an analogous argument. We
suppose that 1—/? is 'small'. Let
6 = 1-P, z = exp(80_1), c=Y\p, fi = \-c~2,
&r={xe(c,c + \):([xn],c)>\}.
Clearly 3F<^ £f. Indeed, if jce J*, then [jc"] takes on at most n(z) prime values.
We pick € so that if n is so large that 2""<€, then
This is possible for all sufficiently small 0, since it requires
log2>- + -
and we have c = exp(exp(80~ *)(1 + 0(1))) as 0 -► 0.
Let v = «-1. We now define a sequence ^ of sets by: <Sl = [c,c + fi], and, for
«^2 when 2~n^rj,
&n = {x:xe[mv,(ra+fi)v]<= ^„_x for some m with (m,c)> 1}.
When 2~"<rj, we impose the extra condition on ^„:
[wv,(w + //)v]n^-# 0=>^)<2-"_1.
Proof of Theorem 10.8 277
We note that, by our construction, each ^ consists of a finite number of closed
intervals, and so is a closed set. Also, clearly
00 00
f| ^„cj%and f] &nnJj=0 for ally.
n=0 n=0
It remains to show that the &n are non-empty, for then their intersection will
be non-empty yet not covered by the J}, which will complete the proof as in
the previous sections. We shall demonstrate that there are in fact
^(c — 3)n-1(l— jQy1 closed intervals [mv,(m + n)v] contained in ^„, and this will
complete the proof.
We have satisfied our requirement for n = \. First suppose €<2~n~1. Now we
note that an interval [wv,(w + /i)v] has length between
C" and H
{m+\)n mn
by the mean-value theorem. Also, if [M1Kn+1), (M+n)1,in+1)] c [wv, (w + aOv], then
M^cm. It follows that each interval in (§n gives rise to
^cmjn + l) ^^^
^ n(m + \) ^
whole intervals in ^n+1 (the final inequality was obtained by noting that m^c,
and trivially n + \>n, c2>c2 — 1; the first inequality was obtained by noting that
at most two intervals lie partly in, partly outside the old interval)—except that
we must count only those M with (m,c)>\. The number of intervals in &n+l
arising is thus at least c — 3 — q>(c). By our construction, <p(c)^clog-1z = c0/8.
Hence the number of intervals m&n+1 arising from each interval in <§n is at least
(c-3)(l-i0). Thus, for all n with 2~n^£, we have ^(c-3)',"1(l-^)',"1
intervals in ^„.
Now, when 2~1 ~n <€ we can repeat the above argument, except that we must
substract the number of intervals which intersect those Ji for which
2~n~2^X(Jj)<2~n~1. From (10.1.1) the number of intervals with this length is
<{2n+2f. The number of intervals [M1/(n+1),(M+/i)1/(n+1)] which intersect these
is thus
2)p
^/c(m + l)(>, + l)2...1+2\20l+
^2c(m + l)(ft+l)2_e(w+1) [ 2(n+B)+1
a
<2c((c + \)n+l)(l-id)n+1(\-5/c)n + 1—-+2
32c
278
Hausdorff dimension of exceptional sets
(using (10.5.1))
<i6{\-W\c-2>)\
We therefore deduce that >(1— \9)\c — 3)" intervals survive, as required to
complete the proof. In the above we noted that if 0 is sufficiently small, then
d
81
2n+3<^-(c-3)n(l-W1
for all n.
□
NOTES
W.M. Schmidt [237] proved that if the numbers 2,3,... are partitioned into two
disjoint classes 01, ¥ such that if re01, set? then (logr/logs)£Q, then there are
numbers which are entirely normal to all bases in 01 but to no base in £f (see also
the earlier result of Cassels [59] where ^ = {3}). Pollington [223] showed that the
Hausdorff dimension of such numbers is one. To do this he needed a result of
Schmidt [238] on sums of Riesz products:
N-l oo
i n
n=lk=K+l
COS
(?0
^2JV1"a(r's),
where
a(r,j)>0if/^5K, !^£Q.
log s
Such expressions occur elsewhere in problems involving normal numbers ([50],
[51]). For example, in [51] it is shown that with sets 01, Sf as above, every real
number is the sum of two numbers normal to every base in 01 but to no base in Sf.
G. Wagner [274] showed how to construct uncountable rings of numbers which,
with the number zero excepted, are normal to one base but not to another.
Schmidt [241] and Pollington [221] have investigated hybrid problems
involving both normality and Diophantine approximation properties.
Pollington [222] has produced a stronger result than Theorem 10.3, namely
that for a lacunary sequence an the set of a for which the sequence {aa„} is not
dense in [0,1) has Hausdorff dimension 1. R.C. Baker [16], [17] has generalized
Theorem 10.4 to give bounds on the Hausdorff dimension of the set Sq of a for
which
limsupZ)N(atfn)iV«"1>0
N-oo
Notes
279
for 0<q<? when an = 0(nc), obtaining
dirndl-^^.
q c + q
In proving their A>dimensional form of Theorem 10.6, Borosh and Fraenkel
appealed to a general result of Moran [197] which is unnecessary in our context.
Theorem 10.7 has not been published previously, although it has been referred to
in the literature (see page 204 of [195]): it was proved by the author and R.C. Baker
in 1992. Theorem 10.8 also has not previously appeared. In view of Theorem 10.9
it would be interesting to have other results on the Hausdorff dimension of the
exceptional sets in Chapter 8, perhaps relating them to the growth of the
sequences.
Regarding other results, we listed some references at the end of Chapter 9 for
the dimension of sets connected with approximation on manifolds. Also, there are
results on the dimension of the set of a whose partial quotients are restricted to
certain sets ([132], [133]). See also [1] for an approach using ergodic theory.
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Index
arithmetic progressions 43, 96, 168-9
asymptotic density, see lower
asymptotic density, upper
asymptotic density
asymptotic formulae 10-11, 13, 60-4,
94-6, 104-5, 109-12, 115, 169,
189-92, 217-20, 242-3
Baker, A. 242, 244
Baker, R.C. 140-2, 163, 188, 201,
213, 220, 245, 278-9
Balog, A. 216
Beck, J. 157-8, 161
Berkes, I. 139-40
Bernik, V.I. 261
Bernstein, F. 25, 97
Besicovitch, A.S. 264
Beurling, A. 127
Bingham, N. 18
Borel-Cantelli lemmas
first 8, 9, 11, 21, 88, 104, 110, 220,
231, 246, 256, 259-60
second 9-10, 22
Borel, E. 1, 3, 5, 7-9, 11, 25, 97;
see also Borel-Cantelli lemmas
Borosh, I. 279
Brun-Titchmarsh inequality 181, 221,
238
Burger, E.B. 59
Cassels, J.W.S. 6, 28-30, 43, 98,
138, 142, 188, 191-2, 278
Catlin, P.A. 28-9
Champernowne, D.G. 23
Chebyshev, P.L. 231
Chen, W.W.L. 161
Cochrane, T. 161
continued fractions 25, 43, 51, 96-7,
112-15,119,144-5,166,168,183,265-6
Corput, J.G. van der 220
Davenport-Erdos-LeVeque
criterion 134
Davenport, H. 51, 82, 134
Deshouillers, J.-M. 216
dimension, see Hausdorff dimension
Diophantine approximation
non-integer sequences 187-214
in one dimension 24-59, 60-1, 66,
94-7, 139, 264-5
on manifolds 93, 241-61
with restricted numerator and
denominator 164-86, 189, 266
in two or more dimensions 63-7,
95, 265
Dirichlet's theorem 24, 26, 51,158,196,
228, 241
discrepancy
definitions 120, 151-2, 157
isotropic 152, 162
logarithmic 130-1
lower bounds 121,130,138-9,145,158
upper bounds 121, 129, 137-8, 142,
145-6, 154, 156, 157, 159-60
Dodson, M.M. 261
Duffin and Schaeffer
conjecture 27, 29, 34, 53, 60, 66,150,
171, 265
in higher dimensions 65-6
results 27, 35, 37-44, 51-3, 96
296
Index
Duffin, R.J., see Duffin and Schaeffer
Dyer, T. 62-3
Eggleston, H.G. 263
Ennola, V. 64
Erdos, P. 27-8, 44, 46-7, 94, 138, 142,
147, 161, 163, 263
see also Erdos-Turan theorem,
Erdos-Turan-Koksma theorem
Erdos-Turan-Koksma theorem
153-4, 157
Erdos-Turan theorem 126, 129-30,
131, 142, 144, 159, 161
ergodic theory 17-18,23,31-4,119,279
essential and inessential domains 246,
249, 254, 259-60
exponential sums 126-31, 132, 134-8,
142-3, 152, 159, 174, 183, 185, 220
222-3, 240, 269-70
Farey sequence 161
Fiedler, H. 138
Fogels, E. 240
Fourier analysis 123, 126-31, 146-7,
152-3, 173, 198-9, 226, 243
Fraenkel, E.S. 279
Gal, I.S. 13, 62, 79, 85
Galambos, J. 23
Gallagher, P.X. 28, 29, 50, 64, 67, 115
GCD sums 53-8, 60-93, 147
generalized 208-11
Halasz, G. 121
Hall, R.R. 240
Hardy, G.H. 4, 32, 161
Hartman, G. 43
Hausdorff dimension 23, 190, 262-79
Heilbronn, H. 240
Hildebrand, A. 77
Hlawka, A. 161-2
Hunt, R.A. 141-2
Hurwitz, A. 24
Huxley, M.N. 220, 234, 240
iterated logarithm, see law of the
iterated logarithm
Jagerman, D.L. 161
Jarnik, V. 264
Jurkat, W. 138
Kargaev, P. 58
Khintchine, A. 18, 25, 27, 58-60, 67,
96-7, 145, 148
Kleinbock, D.Y. 261
Kloosterman sums 183
Knopp, K. 16
Koksma, J.F. 13, 62,124,138,142,146,
161, 192
Kolmogorov's theorem 18
Korner, O. 138
Krause, J.M. 161
Kuipers, L. 4, 161
lacunary sequences 54, 61, 138-9, 147,
189, 218-19, 234, 263-4, 266
Larcher, G. 162
law of the iterated logarithm 18-22, 61,
138
Lebesgue's density theorem, see points
of metric density
Leitman, D. 216
LeVeque, W.J. 60, 63, 69, 134
Levy, P. 96
Littlewood, J.E. 161
logarithmic discrepancy 130-1
lower asymptotic density 41-2, 95-6,
164, 168-70
Mahler, K. 23, 241-2, 244
Margulis, G.A. 261
Marstrand, J.M. 146, 148, 163
Mersenne numbers 215
Mertens' prime-number theorem 150,
225, 237
Mills, W.H. 220
Montgomery, H.L. 141
Moran, P.A.P. 279
Index
297
Nair, R. 157
Niederreiter, H. 4, 161
Niven, I. 4
normal numbers 1-23, 122-3, 263,
266-7, 278
prime-normal 218
overlap estimates 38-40, 44, 50,
55-6, 69-70, 79, 89, 91-2, 97-8,
101-2, 103, 109, 117-18, 150,
170-1, 173, 176, 180, 193-4, 232-9,
245
Philipp, W. 13, 61, 119, 139-40, 188
Piatetski-Shapiro, I. 215
Pillai, S.S. 3
points of metric density 17, 30-1, 34-5,
224
Pollington, A.D. 37, 44, 66, 157, 278
prime-normal numbers 218
primes 27, 32-4, 43-8, 52, 54-5, 59,
148-9, 166, 168, 170, 180-5, 189,
215-40, 266, 275-6
PV numbers 143, 220
Rademacher, H. 13
Raikov, D.J. 148
Riemann hypothesis 161
Riemann integration 123-5, 146-7,
161-2
Riesz, F. 23
Riesz product 278
Rusza, I.Z. 161
Rynne, B.P. 265
Schaeffer, A.C.,
see Duffin and Schaeffer
Schmidt, W.M. 13, 51, 63, 93-119,
121, 155-6, 161, 175, 206, 244, 278
sieve methods 44, 180-3, 195, 221-2,
229
Sprindzuk, V.G. 242, 244, 246
square-free integers 43, 96, 168-9,
177-9, 182
Strauch, O. 44, 58
sums of two squares 43, 95, 170, 240
Szusz, P. 43, 96, 153
Taylor, SJ. 263
Tenenbaum, G. 77
Titchmarsh, E.C. 240, see also
Brun-Titchmarsh inequality
transcendental numbers 190, 241-2
Turan, P., see Erdos-Turan theorem,
Erdos-Turan-Koksma theorem
uniform distribution 120-63
upper asymptotic density 41
Vaaler, J.D. 27-29, 44, 47, 50, 57, 141
Vaughan, R.C. 37, 44, 66
Vilchinskii, V.T. 28, 87, 186
Vinogradov, I.M. 126, 264
Wagner, G. 278
Walfisz, A. 58
Wall, D.D. 122
Weil, A. 183
Weyl criterion 126, 129, 130, 161, 269
Weyl, H. 10-11, see also Weyl criterion
Wolke, D. 216
Wright, E.M. 4, 32
Zaremba, S.F. 161-2
zero-one laws 1, 16-18, 28-36, 50, 87,
119, 164, 166, 171
Zhigljavsky, A. 58
Zuckerman, H.S. 4