ℬ-free sets and dynamics

Abstract

Given B ⊂ N \mathscr {B}\subset \mathbb {N} , let η = η B ∈ { 0 , 1 } Z \eta =\eta _{\mathscr {B}}\in \{0,1\}^{\mathbb {Z}} be the characteristic function of the set F B := Z ∖ ⋃ b ∈ B b Z \mathcal {F}_\mathscr {B}:=\mathbb {Z}\setminus \bigcup _{b\in \mathscr {B}}b\mathbb {Z} of B \mathscr {B} -free numbers. The B \mathscr {B} -free shift ( X η , S ) (X_\eta ,S) , its hereditary closure ( X ~ η , S ) (\widetilde {X}_\eta ,S) , and (still larger) the B \mathscr {B} -admissible shift ( X B , S ) (X_{\mathscr {B}},S) are examined. Originated by Sarnak in 2010 for B \mathscr {B} being the set of square-free numbers, the dynamics of B \mathscr {B} -free shifts was discussed by several authors for B \mathscr {B} being Erdös; i.e., when B \mathscr {B} is infinite, its elements are pairwise coprime, and ∑ b ∈ B 1 / b > ∞ \sum _{b\in \mathscr {B}}1/b>\infty : in the Erdös case, we have X η = X ~ η = X B X_\eta =\widetilde {X}_\eta =X_{\mathscr {B}} . It is proved that X η X_\eta has a unique minimal subset, which turns out to be a Toeplitz dynamical system. Furthermore, a B \mathscr {B} -free shift is proximal if and only if B \mathscr {B} contains an infinite coprime subset. It is also shown that for B \mathscr {B} with light tails, i.e., d ¯ ( ∑ b > K b Z ) → 0 \overline {d}(\sum _{b>K}b\mathbb {Z})\to 0 as K → ∞ K\to \infty , proximality is the same as heredity. For each B \mathscr {B} , it is shown that η \eta is a quasi-generic point for some natural S S -invariant measure ν η u _\eta on X η X_\eta . A special role is played by subshifts given by B \mathscr {B} which are taut, i.e., when δ ( F B ) > δ ( F B ∖ { b } ) \boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}})>\boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}\setminus \{b\}}) for each b ∈ B b\in \mathscr {B} ( δ \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 B \mathscr {B} in terms of the support of ν η u _\eta are given. Moreover, for any B \mathscr {B} there exists a taut B ′ \mathscr {B}’ with ν η B = ν η B ′ u _{\eta _{\mathscr {B}}}=u _{\eta _{\mathscr {B}’}} . For taut sets B , B ′ \mathscr {B},\mathscr {B}’ , it holds that X B = X B ′ X_\mathscr {B}=X_{\mathscr {B}’} if and only if B = B ′ \mathscr {B}=\mathscr {B}’ . For each B \mathscr {B} , it is proved that there exists a taut B ′ \mathscr {B}’ such that ( X ~ η B ′ , S ) (\widetilde {X}_{\eta _{\mathscr {B}’}},S) is a subsystem of ( X ~ η B , S ) (\widetilde {X}_{\eta _{\mathscr {B}}},S) and X ~ η B ′ \widetilde {X}_{\eta _{\mathscr {B}’}} is a quasi-attractor. In particular, all invariant measures for ( X ~ η B , S ) (\widetilde {X}_{\eta _{\mathscr {B}}},S) are supported by X ~ η B ′ \widetilde {X}_{\eta _{\mathscr {B}’}} . Moreover, the system ( X ~ η , S ) (\widetilde {X}_\eta ,S) is shown to be intrinsically ergodic for an arbitrary B \mathscr {B} . A description of all probability invariant measures for ( X ~ η , S ) (\widetilde {X}_\eta ,S) is given. The topological entropies of ( X ~ η , S ) (\widetilde {X}_\eta ,S) and ( X B , S ) (X_\mathscr {B},S) are shown to be the same and equal to d ¯ ( F B ) \overline {d}(\mathcal {F}_\mathscr {B}) ( d ¯ \overline {d} stands for the upper density). Finally, some applications in number theory on gaps between consecutive B \mathscr {B} -free numbers are given, and some of these results are applied to the set of abundant numbers.