Tautness for sets of multiples and applications to ${\mathcal B}$-free dynamics
Volume 247 / 2019
Abstract
For any set $\mathcal B\subseteq\mathbb N=\{1,2,\dots\}$ one can define its set of multiples $\mathcal M_\mathcal B:=\bigcup_{b\in\mathcal B}b\mathbb Z$ and the set of $\mathcal B$-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 in Sarnak’s lectures) 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 $\eta$:
$\bullet$ The measure $\nu_\eta$ has full topological support in $X_\eta$.
$\bullet$ If $X_\eta$ is proximal, i.e. if the one-point set $\{\dots000\dots\}$ 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 coordinatewise product $w\cdot x$ belongs to $X_\eta$.
This strengthens two results of Dymek et al. (2018) which need the stronger assumption that $\mathcal B$ has light tails for the same conclusions.
