Distribution Modulo One and Diophantine Approximation

A treatment of cutting-edge research on the distribution modulo one of sequences and related topics, with exercises and open problems.

Yann Bugeaud (Author)

9780521111690, Cambridge University Press

Hardback, published 5 July 2012

316 pages, 70 exercises
23.3 x 15.7 x 2.1 cm, 0.59 kg

"The reader may learn a lot from this book about various techniques used i this subject over many years (e.g., classical and metrical Diophantine approximation, combinatorics on words) and, in addition, find the proofs of some very recent results." - Arturas Dubickas, Mathematical Reviews

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.

Subject Areas: Combinatorics & graph theory [PBV], Number theory [PBH], Mathematics [PB]