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 "naïvely expected" maximal equicontinuous factor if and only if the interior of the window is aperiodic.

References in corpus (2)

• Dynamics on the graph of the torus parametrisation
• Ergodic properties of visible lattice points

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