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.
1. Distribution modulo one; 2. On the fractional parts of powers of real numbers; 3. On the fractional parts of powers of algebraic numbers; 4. Normal numbers; 5. Further explicit constructions of normal and non-normal numbers; 6. Normality to different bases; 7. Diophantine approximation and digital properties; 8. Digital expansion of algebraic numbers; 9. Continued fraction expansions and beta-expansions; 10. Conjectures and open problems; A. Combinatorics on words; B. Some elementary lemmata; C. Measure theory; D. Continued fractions; E. Diophantine approximation; F. Recurrence sequences; References; Index.
A treatment of cutting-edge research on the distribution modulo one of sequences and related topics, much of it from the last decade. There are numerous exercises to aid student understanding of the topic, and researchers will appreciate the notes at the end of each chapter, extensive references and open problems.
Yann Bugeaud is a Professor in the Institut de Recherche Mathématique Avancée (IRMA) at the University of Strasbourg, France.
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