Abstract

   There is a Russian tradition of formulating promising open problems during semi-nars with a view to promote research. One of the most famous Moscow seminar is led since the 1950’s by Vladimir Igorevich Arnold. His complete collection of prob- lems, known as “Zadachi Arnolda”, has been recently translated and published in English [1]. One of the most recent problems is concerned with the understanding of [1] what Arnold calls the randomness of arithmetic progressions.
   After making precise how Arnold proposes to measure the randomness of a modular sequence, we show that this measure of randomness takes a simplified form in the case of arithmetic progressions. This simplified expression is then estimated using the methodology of dynamical analysis, which operates with tools coming from dynamical systems theory. We deal with various tools: Dirichlet series, Perron’s formula, transfer operators, bounds `a la Dolgopyat.
   In conclusion, this study shows that modular arithmetic progressions are far from behaving like purely random sequences, according to Arnold’s definition. This is by no mean a surprise since it is difficult to imagine a sequence which would be more predictable than an arithmetic progression: nobody would have ever thought to use it as a device to produce random numbers! However, our result provides a precise
estimate for quantifying this non-randomness, which would have been difficult to obtain with elementary means. Our result can also be viewed as a metric version of the classical two distance theorem.

[1] V. I. Arnold, Arnold’s problems, Springer Phasis, 2004.