Generalized heredity in $\mathcal{B}$-free systems

Abstract

Let $\mathcal{B}\subseteq \mathbb{N}$ be a primitive set. We complement results on heredity of the $\mathcal{B}$-free subshift $X_{\eta }$ from [arxiv:1509.08010] in two directions: In the proximal case we prove that a subshift $X_{\phi }$, which micht be slightly larger than the subshift $X_{\eta }$, is always hereditary. (There is no need to assume that the set $\mathcal{B}$ is taut or even has light tails, but if $\mathcal{B}$ is taut, then $X_{\phi }=X_{\eta }$.) We also generalize the the concept of heredity to the non-proximal (and hence non-hereditary) case by proving that $X_{\phi }$ is always "hereditary away from its unique minimal subsystem" (which is always Toeplitz). Finally we characterize regularity of this Toeplitz subsystem equivalently by the condition $m_H(\overline{\mathrm{int}(W)})=0$, where $W$ ("the window") is a subset of a compact abelian group $H$ canonically associated with the set $\mathcal{B}$, and $m_H$ denotes Haar measure on $H$. Throughout, results from [arxiv:1702.02375] are heavily used.

The statement and the proof of Lemma 4 (current version) were corrected. In view of [arxiv:1802.08309], stronger corrolaries to the main result could be formulated

Cited by in corpus (1)

• Tautness for sets of multiples and applications to $\mathcal{B}$-free dynamics