Distribution Modulo One and Diophantine Approximation

318 Pages·2012·1.42 MB·English

by  Yann Bugeaud

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193 Distribution Modulo One and
Diophantine Approximation
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Distribution Modulo One and
Diophantine Approximation
YANN BUGEAUD
Universit´e de Strasbourg
cambridge university press
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(cid:2)c YannBugeaud2012
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Firstpublished2012
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Library of Congress Cataloguing in Publication data
Bugeaud,Yann,1971–
Distributionmodulooneanddiophantineapproximation/YannBugeaud.
p. cm.–(Cambridgetractsinmathematics;193)
Includesbibliographicalreferencesandindex.
ISBN978-0-521-11169-0(hardback)
1. Diophantineanalysis. 2. Distributionmoduloone. I. Title.
QA242.B84 2012
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2012013417
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Contents
Preface page ix
Frequently used notation xv
1 Distribution modulo one 1
1.1 Weyl’s criterion 1
1.2 Metrical results 5
1.3 Discrepancy 9
1.4 Distribution functions 11
1.5 The multidimensional case 12
1.6 Exercises 14
1.7 Notes 14
2 On the fractional parts of powers of real numbers 15
2.1 Thue, Hardy, Pisot and Vijayaraghavan 15
2.2 On some exceptional pairs (ξ,α) 22
2.3 On the powers of real numbers close to 1 28
2.4 On the powers of some transcendental numbers 33
2.5 A theorem of Furstenberg 37
2.6 A conjecture of de Mathan and Teuli´e 39
2.7 Exercises 41
2.8 Notes 44
3 On the fractional parts of powers of algebraic
numbers 48
3.1 The integer case 48
3.2 Mahler’s Z-numbers 50
3.3 On the fractional parts of powers of algebraic
numbers 53
3.4 On the fractional parts of powers of Pisot and
Salem numbers 56
v
vi Contents
3.5 The sequence ((cid:2)ξαn(cid:2))n≥1 61
3.6 Constructions of Pollington and of Dubickas 68
3.7 Waring’s problem 72
3.8 On the integer parts of powers of algebraic numbers 73
3.9 Exercises 73
3.10 Notes 74
4 Normal numbers 78
4.1 Equivalent definitions of normality 79
4.2 The Champernowne number 86
4.3 Normality and uniform distribution 89
4.4 Block complexity and richness 91
4.5 Rational approximation to Champernowne-type
numbers 92
4.6 Exercises 96
4.7 Notes 96
5 Further explicit constructions of normal and non-
normal numbers 102
5.1 Korobov’s and Stoneham’s normal numbers 102
5.2 Absolutely normal numbers 111
5.3 Absolutely non-normal numbers 112
5.4 On the random character of arithmetical constants 114
5.5 Exercises 116
5.6 Notes 116
6 Normality to different bases 118
6.1 Normality to a prescribed set of integer bases 118
6.2 Normality to non-integer bases 123
6.3 Ontheexpansionsofarealnumbertotwodifferent
bases 131
6.4 Ontherepresentationofanintegerintwodifferent
bases 134
6.5 Exercises 135
6.6 Notes 135
7 Diophantine approximation and digital properties 139
7.1 Exponents of Diophantine approximation 140
7.2 Prescribing simultaneously the values of all the
exponents v 144
b
7.3 Badly approximable numbers to integer bases 147
7.4 Almost no element of the middle third Cantor set
is very well approximable 153
Contents vii
7.5 Playing games on the middle third Cantor set 156
7.6 Elements of the middle third Cantor set with
prescribed irrationality exponent 158
7.7 Normal and non-normal numbers with prescribed
Diophantine properties 161
7.8 Hausdorff dimension of sets with missing digits 163
7.9 Exercises 165
7.10 Notes 166
8 Digital expansion of algebraic numbers 170
8.1 A transcendence criterion 171
8.2 Block complexity of algebraic numbers 173
8.3 Zeros in the b-ary expansion of algebraic numbers 176
8.4 Number of digit changes in the b-ary expansion of
algebraic numbers 182
8.5 On the b-ary expansion of e and some other
transcendental numbers 184
8.6 Onthedigitsofthemultiplesofanirrationalnumber 185
8.7 Exercises 189
8.8 Notes 192
9 Continued fraction expansions and β-expansions 195
9.1 Normal continued fractions 195
9.2 Onthecontinuedfractionexpansionofanalgebraic
number 201
9.3 On β-expansions 206
9.4 Exercises 209
9.5 Notes 210
10 Conjectures and open questions 214
Appendix A Combinatorics on words 223
Appendix B Some elementary lemmata 231
Appendix C Measure theory 234
Appendix D Continued fractions 241
Appendix E Diophantine approximation 246
Appendix F Recurrence sequences 253
References 257
Index 299