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Uspekhi Mat. Nauk, 2008, Volume 63, Issue 2(380), Pages 5–20 (Mi umn9191)  

This article is cited in 14 scientific papers (total in 14 papers)

To what extent are arithmetic progressions of fractional parts stochastic?

V. I. Arnol'd

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: 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).

DOI: https://doi.org/10.4213/rm9191

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English version:
Russian Mathematical Surveys, 2008, 63:2, 205–220

Bibliographic databases:

Document Type: Article
MSC: Primary 11B25; Secondary 11A55, 11K45
Received: 10.12.2007

Citation: V. I. Arnol'd, “To what extent are arithmetic progressions of fractional parts stochastic?”, Uspekhi Mat. Nauk, 63:2(380) (2008), 5–20; Russian Math. Surveys, 63:2 (2008), 205–220

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