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Tautness for sets of multiples and applications to $\mathcal B$-free dynamics

For any set $\mathcal B\subseteq\mathbb N=\{1,2,\dots\}$ one can define its \emph{set of multiples} $\mathcal M_{\mathcal B}:=\bigcup_{b\in\mathcal B}b\mathbb Z$ and the set of \emph{$\mathcal B$-free numbers} $\mathcal F_{\mathcal B}:=\mathbb Z\setminus\mathcal M_{\mathcal B}$. Tautness of the set $\mathcal B$ is a basic property related to questions around the asymptotic density of $\mathcal M_{\mathcal B}\subseteq\mathbb Z$. From a dynamical systems point of view (originated by Sarnak) one studies $η$, the indicator function of $\mathcal F_{\mathcal B}\subseteq\mathbb Z$, its shift-orbit closure $X_η\subseteq\{0,1\}^{\mathbb Z}$ and the stationary probability measure $ν_η$ defined on $X_η$ by the frequencies of finite blocks in $η$. In this paper we prove that tautness implies the following two properties of $η$: (1) The measure $ν_η$ has full topological support in $X_η$. (2) If $X_η$ is proximal, i.e. if the one-point set $\{\dots000\dots\}$ is contained in $X_η$ and is the unique minimal subset of $X_η$, then $X_η$ is hereditary, i.e. if $x\in X_η$ and if $w$ is an arbitrary element of $\{0,1\}^{\mathbb Z}$, then also the coordinate-wise product $w\cdot x$ belongs to $X_η$. This strengthens two results from [Bartnicka et al. 2015] which need the stronger assumption that $\mathcal B$ has light tails for the same conclusions.