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

Abstract

Let $B\subset \mathbb{N}$ and let $η\in \{0,1\}^\mathbb{Z}$ be the characteristic function of the set $F_B:=\mathbb{Z}\setminus\bigcup_{b}b\mathbb{Z}$ of B-free numbers. Consider $(S,X_η)$, where $X_η$ is the closure of the orbit of $η$ under the left shift S. When $B=\{p^2 : p\in P\}$, $(S,X_η)$ 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 $X_η\subsetneq X_B:=\{x\in \{0,1\}^\mathbb{Z} : |\text{supp }x\bmod b|\leq b-1 \forall_b\}$. Also, $X_η$ may not be hereditary (heredity means that if $x\in X$ and $y\leq x$ coordinatewise then $y\in X$). We show that $η$ is quasi-generic for a natural measure $ν_η$. We solve the problem of proximality by showing first that $X_η$ has a unique minimal (Toeplitz) subsystem. Moreover B-free system is proximal iff B contains an infinite coprime set. B is taut when $δ(F_B)<δ(F_{B\setminus \{b\} })$ for each b. We give a characterization of taut B in terms of the support of $ν_η$. Moreover, for any B there exists a taut B' with $ν_η=ν_{η'}$. For taut sets B,B', we have B=B' iff $X_B=X_{B'}$. For each B there is a taut B' with $\tilde{X}_{η'}\subset \tilde{X}_η$ and all invariant measures for $(S,\tilde{X}_η)$ live on $\tilde{X}_{η'}$. $(S,\tilde{X}_η)$ is shown to be intrinsically ergodic for all B. We give a description of all invariant measures for $(S,\tilde{X}_η)$. The topological entropies of $(S,\tilde{X}_η)$ 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.