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

Let $B\subset \mathbb{N}$ and let $\eta\in \{0,1\}^\mathbb{Z}$ be the characteristic function of the set $F_B:=\mathbb{Z}\setminus\bigcup_{b}b\mathbb{Z}$ of B-free numbers. Consider $(S,X_\eta)$, where $X_\eta$ is the closure of the orbit of $\eta$ under the left shift S. When $B=\{p^2 : p\in P\}$, $(S,X_\eta)$ was studied by Sarnak. This case + some generalizations, including the case (*) of B infinite, coprime with $\sum_{b}1/b<\infty$, were discussed by several authors. For general B, contrary to (*), we may have $X_\eta\subsetneq X_B:=\{x\in \{0,1\}^\mathbb{Z} : |\text{supp }x\bmod b|\leq b-1 \forall_b\}$. Also, $X_\eta$ may not be hereditary (heredity means that if $x\in X$ and $y\leq x$ coordinatewise then $y\in X$). We show that $\eta$ is quasi-generic for a natural measure $\nu_\eta$. We solve the problem of proximality by showing first that $X_\eta$ has a unique minimal (Toeplitz) subsystem. Moreover B-free system is proximal iff B contains an infinite coprime set. B is taut when $\delta(F_B)<\delta(F_{B\setminus \{b\} })$ for each b. We give a characterization of taut B in terms of the support of $\nu_\eta$. Moreover, for any B there exists a taut B' with $\nu_\eta=\nu_{\eta'}$. For taut sets B,B', we have B=B' iff $X_B=X_{B'}$. For each B there is a taut B' with $\tilde{X}_{\eta'}\subset \tilde{X}_\eta$ and all invariant measures for $(S,\tilde{X}_\eta)$ live on $\tilde{X}_{\eta'}$. $(S,\tilde{X}_\eta)$ is shown to be intrinsically ergodic for all B. We give a description of all invariant measures for $(S,\tilde{X}_\eta)$. The topological entropies of $(S,\tilde{X}_\eta)$ and $(S,X_B)$ are both equal to $\overline{d}(F_B)$. We show that for a subclass of taut B-free systems proximality is the same as heredity. Finally, we give applications in number theory on gaps between consecutive B-free numbers. We apply our results to the set of abundant numbers.