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

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. A dynamical point of view on the set of B-free integers. International Mathematics Research Notices, Oxford University Press (OUP), 2014, pp.doi: 10.1093/imrn/rnu164. ⟨10.1093/imrn/rnu164⟩. ⟨hal-00904678v4⟩



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