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arXiv · 1311.3752

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

El Houcein El Abdalaoui·Mariusz Lemanczyk·Thierry De La Rue
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Abstract

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.

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El Houcein El Abdalaoui, Mariusz Lemanczyk, Thierry De La Rue. 2013-11-23. A dynamical point of view on the set of B-free integers. https://arxiv.org/abs/1311.3752

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