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Probabilistic Diophantine Approximation

József Beck
Publication Date: 
Number of Pages: 
Springer Monographs in Mathematics
[Reviewed by
Darren Glass
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Let \(\alpha\) be an irrational number, and consider the sequence \(\alpha , 2\alpha , 3\alpha ,\dots\) modulo one. It is likely that the reader knows that the sequence is dense in the unit interval \((0,1)\), which is a classical result often attributed to Kronecker. Some readers probably know the stronger statement proven by Bohl, Sierpinski, and Weyl that the sequence is also uniformly distributed on \((0,1)\). However, it turns out that much more can be said about the regularity of this sequence. For example, there is the following result, which this reviewer found very surprising:

Three Distance Theorem: Consider the first \(n\) terms of the sequence \(\alpha , 2\alpha , 3\alpha ,...\) modulo one and label them in increasing order \( 0 < \gamma_1 < \gamma_2 < ... < \gamma_n < 1\) with the additional definitions that \(\gamma_0=0\) and \(\gamma_{n+1}=1\). Then the set \(\{\gamma_{i+1} - \gamma_i\}\) consists of at most three values.

Even more can be said if \(\alpha\) is a quadratic irrational number, using the fact that the continued fraction expansion of \(\alpha\) will have nice properties. In particular, it turns out that under these hypotheses the sequence will exhibit a central limit theorem that looks very analogous to the types of things that one would see in a book on probability theory. It is for this reason that Jozsef Beck almost gave his new book, Probabilistic Diophantine Approximation the subtitle “Randomness of \(\sqrt{2}\).” As he discusses in the introduction, he felt that subtitle would be misleading and ultimately decided on “Randomness in Lattice Point Counting,” which gives a hint as to some of the techniques that he uses through the book. Beck’s book brings together probability theory, algebraic number theory, and combinatorics to forge new ground.

The bulk of the book is dedicated to proving several results, each of which can be described as a Central Limit Theorem (and each of which is too technical to reproduce in this review). The first half of Beck’s book is dedicated to the global aspects (“Randomness of the Irrational Rotation”) and the second half dedicated to local aspects (“Inhomogeneous Pell Inequalities”) of the situation. The book is quite dense in the amount of technical mathematics it covers, but throughout the book, Beck gives motivating examples and describes connections with various areas of mathematics and helps give outlines of proofs before diving into the details. (In fact, my biggest complaint with the book is that I often last track of what the author had actually proved and what he had simply told us he was going to prove).

Beck had previously written a dozen or so research papers on these types of results, and much of the book is dedicated to recapping and expanding on those results, while also filling in the background for readers who may not be experts in all of these different branches of mathematics (in other words, the vast majority of readers). As he writes in his introduction, “‘Algebraists’ and ‘probabilists’ are in fact very different kinds of mathematicians with totally different taste and different intuitions,” and Beck is trying to appeal to both tastes. His goal was to write a book that would be accessible to beginning graduate students in all areas of mathematics, and while this may be overly optimistic given the level of the material I do think the book is accessible to a wide range of mathematicians. In my own experience, while there were a number of occasions where I had to pull out other references to remind/teach myself various pieces of background material, Beck did a nice job of motivating the questions and explaining the answers.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College whose primary mathematical interests include Number Theory, Algebraic Geometry, and Graph Theory. He can be reached at


1 What is "probabilistic" diophantine approximation?

2 Expectation, and its connection with quadratic fields

3 Variance, and its connection with quadratic fields

4 Proving randomness

5 Pell equation, super irregularity and randomness

6 More on randomness