A Dynamical Point of View on the Set of B-Free Integers

作者：El Abdalaoui El Houcein; Lemanczyk Mariusz; de la Rue Thierry^*

来源：International Mathematics Research Notices, 2015, 2015(16): 7258-7286.

DOI：10.1093/imrn/rnu164

Sarnak has recently initiated the study of the Mpbius function and its square, the characteristic function of square-free integers, from a dynamical point of view, introducing the Mobius flow and the square-free flow as the action of the shift map on the respective subshfits generated by these functions. In this paper, we extend the study of the square-free flow 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 obtain the abundance of twin B-free numbers. Moreover, we show that the distribution of patterns in small intervals of the form [N, N + root N) also conforms to the same measure. When elements of B are squares, we introduce a generalization of the Mobius function, and discuss a conjecture of Chowla in this broader context.

• 出版日期2015