$\mathscr {B}$-free sets and dynamics

Abstract:

Given $\mathscr {B}\subset \mathbb {N}$, let $\eta =\eta _{\mathscr {B}}\in \{0,1\}^{\mathbb {Z}}$ be the characteristic function of the set $\mathcal {F}_\mathscr {B}:=\mathbb {Z}\setminus \bigcup _{b\in \mathscr {B}}b\mathbb {Z}$ of $\mathscr {B}$-free numbers. The $\mathscr {B}$-free shift $(X_\eta ,S)$, its hereditary closure $(\widetilde {X}_\eta ,S)$, and (still larger) the $\mathscr {B}$-admissible shift $(X_{\mathscr {B}},S)$ are examined. Originated by Sarnak in 2010 for $\mathscr {B}$ being the set of square-free numbers, the dynamics of $\mathscr {B}$-free shifts was discussed by several authors for $\mathscr {B}$ being Erdös; i.e., when $\mathscr {B}$ is infinite, its elements are pairwise coprime, and $\sum _{b\in \mathscr {B}}1/b<\infty$: in the Erdös case, we have $X_\eta =\widetilde {X}_\eta =X_{\mathscr {B}}$.

It is proved that $X_\eta$ has a unique minimal subset, which turns out to be a Toeplitz dynamical system. Furthermore, a $\mathscr {B}$-free shift is proximal if and only if $\mathscr {B}$ contains an infinite coprime subset. It is also shown that for $\mathscr {B}$ with light tails, i.e., $\overline {d}(\sum _{b>K}b\mathbb {Z})\to 0$ as $K\to \infty$, proximality is the same as heredity.

For each $\mathscr {B}$, it is shown that $\eta$ is a quasi-generic point for some natural $S$-invariant measure $\nu _\eta$ on $X_\eta$. A special role is played by subshifts given by $\mathscr {B}$ which are taut, i.e., when $\boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}})<\boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}\setminus \{b\}})$ for each $b\in \mathscr {B}$ ($\boldsymbol {\delta }$ stands for the logarithmic density). The taut class contains the light tail case; hence all Erdös sets and a characterization of taut sets $\mathscr {B}$ in terms of the support of $\nu _\eta$ are given. Moreover, for any $\mathscr {B}$ there exists a taut $\mathscr {B}’$ with $\nu _{\eta _{\mathscr {B}}}=\nu _{\eta _{\mathscr {B}’}}$. For taut sets $\mathscr {B},\mathscr {B}’$, it holds that $X_\mathscr {B}=X_{\mathscr {B}’}$ if and only if $\mathscr {B}=\mathscr {B}’$.

For each $\mathscr {B}$, it is proved that there exists a taut $\mathscr {B}’$ such that $(\widetilde {X}_{\eta _{\mathscr {B}’}},S)$ is a subsystem of $(\widetilde {X}_{\eta _{\mathscr {B}}},S)$ and $\widetilde {X}_{\eta _{\mathscr {B}’}}$ is a quasi-attractor. In particular, all invariant measures for $(\widetilde {X}_{\eta _{\mathscr {B}}},S)$ are supported by $\widetilde {X}_{\eta _{\mathscr {B}’}}$. Moreover, the system $(\widetilde {X}_\eta ,S)$ is shown to be intrinsically ergodic for an arbitrary $\mathscr {B}$. A description of all probability invariant measures for $(\widetilde {X}_\eta ,S)$ is given. The topological entropies of $(\widetilde {X}_\eta ,S)$ and $(X_\mathscr {B},S)$ are shown to be the same and equal to $\overline {d}(\mathcal {F}_\mathscr {B})$ ($\overline {d}$ stands for the upper density).

Finally, some applications in number theory on gaps between consecutive $\mathscr {B}$-free numbers are given, and some of these results are applied to the set of abundant numbers.