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

Abstract

Sarnak has recently initiated the study of the Möbius function and its square, the characteristic function of square-free integers, from a dynamical point of view, introducing the Möbius 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 |$\mathscr {B}$|-free integers, that is to say integers with no factor in a given family |$\mathscr {B}$| 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 |$\mathscr {B}$|-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 |$\mathscr {B}$|-free numbers. Moreover, we show that the distribution of patterns in small intervals of the form |$[N,N+ \sqrt {N})$| also conforms to the same measure. When elements of |$\mathscr {B}$| are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.