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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 2016, Volume 61, Issue 4, Pages 805–829 (Mi tvp5088)

Ergodic and statistical properties of $\mathscr{B}$-free numbers

M. Avdeevaa, F. Cellarosia, Ya. G. Sinaibc

a Department of Mathematics and Statistics, Queen's University
b Princeton University, Department of Mathematics
c L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences

Abstract: In this survey, we outline several results on the distribution of $B$-free integers and explore a random process naturally associated to them. We show how, notwithstanding the rigid ergodic properties of this process (zero entropy, pure point spectrum, no weak mixing), it exhibits a central limit theorem resembling a theorem by Beck on the circle rotation by a quadratic surd. We explain the connection of the random process to the distribution of $B$-free integers in short intervals, with particular emphasis on their variance and higher moments.

Keywords: $B$-free integers, Möbius function, entropy, correlation functions, central limit theorem.

DOI: https://doi.org/10.4213/tvp5088

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English version:
Theory of Probability and its Applications, 2017, 61:4, 569–589

Bibliographic databases:

Document Type: Article
Language: English

Citation: M. Avdeeva, F. Cellarosi, Ya. G. Sinai, “Ergodic and statistical properties of $\mathscr{B}$-free numbers”, Teor. Veroyatnost. i Primenen., 61:4 (2016), 805–829; Theory Probab. Appl., 61:4 (2017), 569–589

Citation in format AMSBIB
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