Dynamics of $\mathcal B$-free sets: a view through the window

Abstract

Let $\mathcal B$ be an infinite subset of $\{1,2,\dots\}$. We characterize arithmetic and dynamical properties of the $\mathcal B$-free set $\mathcal F_{\mathcal B}$ through group theoretical, topological and measure theoretic properties of a set $W$ (called the window) associated with $\mathcal B$. This point of view stems from the interpretation of the set $\mathcal F_{\mathcal B}$ as a weak model set. Our main results are: $\mathcal B$ is taut if and only if the window is Haar regular; the dynamical system associated to $\mathcal F_{\mathcal B}$ is a Toeplitz system if and only if the window is topologically regular; the dynamical system associated to $\mathcal F_{\mathcal B}$ is proximal if and only if the window has empty interior; and the dynamical system associated to $\mathcal F_{\mathcal B}$ has the "na\"ively expected" maximal equicontinuous factor if and only if the interior of the window is aperiodic.