Geodesic
Ergodic and statistical properties of $\mathscr{B}$-free numbers
Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 4, pp. 805-829

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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. 
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     author = {M. Avdeeva and F. Cellarosi and Ya. G. Sinai},
     title = {Ergodic and statistical properties of $\mathscr{B}$-free numbers},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {805--829},
     publisher = {mathdoc},
     volume = {61},
     number = {4},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2016_61_4_a8/}
}
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M. Avdeeva; F. Cellarosi; Ya. G. Sinai. Ergodic and statistical properties of $\mathscr{B}$-free numbers. Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 4, pp. 805-829. http://geodesic.mathdoc.fr/item/TVP_2016_61_4_a8/