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

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 $\mathcal F_{\mathcal B}$asaweakmodelset\mathrm{.}Ourmainresultsare:$\mathcal B$ 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.