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Distribution Modulo One and Diophantine Approximation

Book Description

This book presents state-of-the-art research on the distribution modulo one of sequences of integral powers of real numbers and related topics. Most of the results have never before appeared in one book and many of them were proved only during the last decade. Topics covered include the distribution modulo one of the integral powers of 3/2 and the frequency of occurrence of each digit in the decimal expansion of the square root of two. The author takes a point of view from combinatorics on words and introduces a variety of techniques, including explicit constructions of normal numbers, Schmidt's games, Riesz product measures and transcendence results. With numerous exercises, the book is ideal for graduate courses on Diophantine approximation or as an introduction to distribution modulo one for non-experts. Specialists will appreciate the inclusion of over 50 open problems and the rich and comprehensive bibliography of over 700 references.

Table of Contents

  1. Cover Page
  2. Half Title
  3. About Editors
  4. Title Page
  5. Copyright Page
  6. Table of Contents
  7. Preface
  8. Frequently used notation
  9. 1 Distribution modulo one
    1. 1.1 Weyl’s criterion
    2. 1.2 Metrical results
    3. 1.3 Discrepancy
    4. 1.4 Distribution functions
    5. 1.5 The multidimensional case
    6. 1.6 Exercises
    7. 1.7 Notes
  10. 2 On the fractional parts of powers of real numbers
    1. 2.1 Thue, Hardy, Pisot and Vijayaraghavan
    2. 2.2 On some exceptional pairs (ξ, α)
    3. 2.3 On the powers of real numbers close to 1
    4. 2.4 On the powers of some transcendental numbers
    5. 2.5 A theorem of Furstenberg
    6. 2.6 A conjecture of de Mathan and Teulié
    7. 2.7 Exercises
    8. 2.8 Notes
  11. 3 On the fractional parts of powers of algebraic numbers
    1. 3.1 The integer case
    2. 3.2 Mahler’s Z-numbers
    3. 3.3 On the fractional parts of powers of algebraic numbers
    4. 3.4 On the fractional parts of powers of Pisot and Salem numbers
    5. 3.5 The sequence (||ξαn)n≥1
    6. 3.6 Constructions of Pollington and of Dubickas
    7. 3.7 Waring’s problem
    8. 3.8 On the integer parts of powers of algebraic numbers
    9. 3.9 Exercises
    10. 3.10 Notes
  12. 4 Normal numbers
    1. 4.1 Equivalent definitions of normality
    2. 4.2 The Champernowne number
    3. 4.3 Normality and uniform distribution
    4. 4.4 Block complexity and richness
    5. 4.5 Rational approximation to Champernowne-type numbers
    6. 4.6 Exercises
    7. 4.7 Notes
  13. 5 Further explicit constructions of normal and non-normal numbers
    1. 5.1 Korobov’s and Stoneham’s normal numbers
    2. 5.2 Absolutely normal numbers
    3. 5.3 Absolutely non-normal numbers
    4. 5.4 On the random character of arithmetical constants
    5. 5.5 Exercises
    6. 5.6 Notes
  14. 6 Normality to different bases
    1. 6.1 Normality to a prescribed set of integer bases
    2. 6.2 Normality to non-integer bases
    3. 6.3 On the expansions of a real number to two different bases
    4. 6.4 On the representation of an integer in two different bases
    5. 6.5 Exercises
    6. 6.6 Notes
  15. 7 Diophantine approximation and digital properties
    1. 7.1 Exponents of Diophantine approximation
    2. 7.2 Prescribing simultaneously the values of all the exponents vb
    3. 7.3 Badly approximable numbers to integer bases
    4. 7.4 Almost no element of the middle third Cantor set is very well approximable
    5. 7.5 Playing games on the middle third Cantor set
    6. 7.6 Elements of the middle third Cantor set with prescribed irrationality exponent
    7. 7.7 Normal and non-normal numbers with prescribed Diophantine properties
    8. 7.8 Hausdorff dimension of sets with missing digits
    9. 7.9 Exercises
    10. 7.10 Notes
  16. 8 Digital expansion of algebraic numbers
    1. 8.1 A transcendence criterion
    2. 8.2 Block complexity of algebraic numbers
    3. 8.3 Zeros in the b-ary expansion of algebraic numbers
    4. 8.4 Number of digit changes in the b-ary expansion of algebraic numbers
    5. 8.5 On the b-ary expansion of e and some other transcendental numbers
    6. 8.6 On the digits of the multiples of an irrational number
    7. 8.7 Exercises
    8. 8.8 Notes
  17. 9 Continued fraction expansions and β-expansions
    1. 9.1 Normal continued fractions
    2. 9.2 On the continued fraction expansion of an algebraic number
    3. 9.3 On β-expansions
    4. 9.4 Exercises
    5. 9.5 Notes
  18. 10 Conjectures and open questions
  19. Appendix A Combinatorics on words
  20. Appendix B Some elementary lemmata
  21. Appendix C Measure theory
  22. Appendix D Continued fractions
  23. Appendix E Diophantine approximation
  24. Appendix F Recurrence sequences
  25. References
  26. Index