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Distribution Modulo One and Diophantine Approximation

Yann Bugeaud
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Tracts in Mathematics 193
[Reviewed by
Underwood Dudley
, on

A sequence of real numbers is uniformly distributed (mod 1) if its fractional parts distribute themselves along the unit interval as if they were random. That is, in the limit, the proportion of them in an interval of length d is d.

In 1916 Hermann Weyl gave his criterion for uniform distribution (mod 1): a sequence {xn} is u. d. (mod 1) if and only if

 limit condition

for positive integers r, where e(a) = e2πia. It implies that {} is u. d. (mod 1) if and only if α is irrational and is the basis for many other results.

The author says that the main questions that motivated his book are

Is there a transcendental real number α such that ‖αn‖ tends to 0 as n tends to infinity?
Is the sequence of fractional parts of {(3/2)n}, n ≥ 1, dense in the unit interval?
What can be said on the digital expansion of an irrational algebraic number?

The book contains ten chapters that are largely independent of each other, ending with a chapter of conjectures and open questions. See the table of contents for what is in the various chapters.

Many results, many of them recent, are given. Each chapter ends with interesting notes. There are exercises (no hints or answers provided) that should not be attempted by the faint of heart. The list of references contains seven hundred and fifty-one items.

This is not a book for the general mathematical reader, but specialists should find it of great value. As always with Cambridge Mathematical Tracts, the book is a delight to the eye. If there are any misprints I didn’t notice them.

In 1964 Woody Dudley proved that though {cos } is not uniformly distributed (mod 1) for almost all θ, {f(n)cos } is, where f is any function such that f(n) goes to infinity with n, no matter how slowly. It is his best theorem.

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