To what extent are arithmetic progressions of fractional parts stochastic?

V. I. Arnol'd

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 63 (2008) no. 2, pp. 205-220

Résumé

For the sequence of residues of division of $n$ members of an arithmetic progression by a real number $N$, it is proved that the Kolmogorov stochasticity parameter $\lambda_n$ tends to 0 as $n$ tends to infinity when the progression step is commensurable with $N$. In contrast, for the case when the step is incommensurable with $N$, examples are given in which the stochasticity parameter $\lambda_n$ not only does not tend to 0, but even takes some arbitrary large values (infrequently). Too small and too large values of the stochasticity parameter both indicate a small probability that the corresponding sequence is random. Thus, long arithmetic progressions of fractional parts are apparently much less stochastic than for geometric progressions (which provide moderate values of the stochasticity parameter, similar to its values for genuinely random sequences).