Distribution Modulo One and Diophantine ApproximationThis 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. |
Contents
On the fractional parts of powers of real numbers | 15 |
On the fractional parts of powers of algebraic | 48 |
Normal numbers | 78 |
Further explicit constructions of normal and non | 102 |
Normality to different bases | 118 |
Diophantine approximation and digital properties | 139 |
6 | 141 |
7 | 161 |
8 | 172 |
4 | 209 |
Conjectures and open questions | 218 |
Appendix B Some elementary lemmata | 231 |
Continued fractions | 241 |
Appendix F Recurrence sequences | 253 |
299 | 273 |
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Common terms and phrases
Acta Arith algebraic integer algebraic number b-ary expansion block Bugeaud Chapter 9 Cited in Appendix Cited in Chapter Consequently construction continued fraction expansion converges coprime coprime integers deduce defined DEFINITION denote digits Diophantine approximation Dubickas Erdős Ergodic Ergodic Theory established EXERCISE exists a positive finite Galois conjugates given Hausdorff dimension implies infinite word integer base irrational number irrational real number irrationality exponent Lebesgue measure Lemma lim sup limit point Mahler Math middle third Cantor Nombres non-zero real number normal numbers normal to base number of degree Number Theory partial quotients Pisot Pisot number polynomial positive integer positive real number prime number PROBLEM proof of Theorem proved rational number recurrence sequence result Salem number satisfies Section set of real simply normal Sturmian word Subspace Theorem tends to infinity Theorem 1.2 third Cantor set ultimately periodic uniformly distributed modulo