Distribution modulo one and diophantine approximation 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 Schmidts 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