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

Abstract

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 $\eta$, the indicator function of ${\mathcal{F}}_{\mathcal{B}}\subseteq \mathbb{Z}$, its shift-orbit closure $X_{\eta }\subseteq \{0,1{\}}^{\mathbb{Z}}$ and the stationary probability measure ${\nu }_{\eta }$ defined on $X_{\eta }$ by the frequencies of finite blocks in $\eta$. In this paper we prove that tautness implies the following two properties of $\eta$: (1) The measure ${\nu }_{\eta }$ has full topological support in $X_{\eta }$. (2) If $X_{\eta }$ is proximal, i.e. if the one-point set $\{\dots 000\dots \}$ is contained in $X_{\eta }$ and is the unique minimal subset of $X_{\eta }$, then $X_{\eta }$ is hereditary, i.e. if $x\in X_{\eta }$ and if $w$ is an arbitrary element of $\{0,1{\}}^{\mathbb{Z}}$, then also the coordinate-wise product $w\cdot x$ belongs to $X_{\eta }$. This strengthens two results from [Bartnicka et al. 2015] which need the stronger assumption that $\mathcal{B}$ has light tails for the same conclusions.

Minor correction to Corollary 1, main results unchanged, typos corrected, references updated. An erratum to the version published in Studia Math. is to appear

References in corpus (2)

• Dynamics on the graph of the torus parametrisation
• Generalized heredity in $\mathcal{B}$-free systems