Ergodic properties of square-free numbers

By F. Cellarosi, Y. Sinai. 2011

70 citations recorded by the source corpus.

We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fc58dc1711632ce25ae6ccf5c8414aa18304685d

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In Mathematical Physics

Citations in the corpus

Dynamics on the graph of the torus parametrization (2015, methodology): “With B = {p2 : p prime}, the set VB are the square-free integers [51, 45, 17].”
The logarithmic Sarnak conjecture for ergodic weights (2017, background): “3 Note that all three possibilities can occur, for example it is known that the Furstenberg system of μ2 (called the square-free system) is an ergodic odometer [11], and conditional to the Chowla conjecture it is known that the Furstenberg system of the Liouville function λ is isomorphic to a Bernou”
Ergodic properties of visible lattice points (2015, background): “It is well known that Kμ = (G,Z 2, μ) has the same pure point dynamical spectrum as (XV ,Z 2, ν); compare [13] for the details in the case of the square-free integers.”
Sarnak’s Conjecture: What’s New (2017, background): “On the other hand, pXμ2 , Sq has positive topological entropy [6, 136, 148] and μ(2) P Xμ2 is completely deterministic, see [3, 33].”
The Chowla and the Sarnak conjectures from ergodic theory point of view (2014, background): “z2 is quasi-generic for along (N k), i.e. we have (19) N k;z2 := 1 N k X nN k Snz2 ! k!1 2P S(X z2): Remark 4.4. In the classical situation z= , z2 is generic for the Mirsky measure [18], cf. [4,22]. Moreover, the Mirsky measure on X z2 has full topological support, cf. (50). In a more general fra”
On invariant measures for B-free systems (2014, unknown): “· bk, k ≥ 1, see [1], [4], [13].”