A dynamical point of view on the set of B-free integers

El Houcein El Abdalaoui (1) , Mariusz Lemanczyk (2) , Thierry de La Rue (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 Faculty of Mathematics and Computer Science

We extend the study of the square-free flow, recently introduced by Sarnak, to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the B-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we get the abundance of twin B-free integers. Moreover, we show that the distribution of patterns in small intervals also conforms to the same measure. When elements of B are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.

Mots clés: topological and Kolmogorov entropy.; B-free integers; Möbius function; Möbius flow; square-free flow; Chowla conjecture; topological and Kolmogorov entropy

Domaines: Systèmes dynamiques [math.DS]; Théorie des nombres [math.NT]

El Houcein El Abdalaoui, Mariusz Lemanczyk, Thierry de La Rue. A dynamical point of view on the set of B-free integers. International Mathematics Research Notices, 2014, pp.doi: 10.1093/imrn/rnu164. ⟨10.1093/imrn/rnu164⟩. ⟨hal-00904678v4⟩