Distribution Modulo One and Diophantine Approximation

Abstract

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.

Around the Littlewood conjecture in Diophantine approximation

“…An alternative proof of Theorem 10 was given by Boshernitzan [8] and is reproduced in the monograph [13].…”

A polynomial-time algorithm for computing absolutely normal numbers

“…Although it seems more demanding, this last condition is equivalent to requiring that just R be simply normal to the bases b n , for every n ě 1. A proof can be read in [4]. ă pk{b´εk{2qpb´1q k´k{b`εk{2 " 1´ε b{2 1´1{b`ε{2 ă 1´ε b{2, pusing ε ď 1{bq ă e´b ε{2 .…”

“…We are left with the question of whether the trade-off between rate of computation and rate of convergence to normal is an inherent aspect of any computation of an absolutely normal number or an artifact of our construction. There are known limits on the rate of convergence to normality and there are examples that are nearly optimal [4,Chapter 4].…”

The number of large prime factors of integers and normal numbers

“…To name only a few, let us mention Nakai and Shiokawa [15], Madritsch, Thuswaldner and Tichy [14] and finally Vandehey [17]. More examples of normal numbers as well as numerous references can be found in the recent book of Bugeaud [1]. In a series of papers, we also constructed large families of normal numbers using the distribution of the values of P (n), the largest prime factor function (see [6], [7], [8] and [9]).…”