Keller G (2019)
Publication Language: English
Publication Status: Accepted
Publication Type: Journal article, Online publication
Future Publication Type: Journal article
Publication year: 2019
Book Volume: 247
Pages Range: 205-216
URI: https://arxiv.org/abs/1802.08309
DOI: 10.4064/sm180305-9-4
For any set ⊆ℕ={1,2,…} one can define its emph{set of multiples} :=⋃b∈bℤ and the set of emph{-free numbers} :=ℤ∖. Tautness of the set is a basic property related to questions around the asymptotic density of ⊆ℤ. From a dynamical systems point of view (originated by Sarnak) one studies η, the indicator function of ⊆ℤ, its shift-orbit closure Xη⊆{0,1}ℤ and the stationary probability measure νη defined on Xη by the frequencies of finite blocks in η. In this paper we prove that tautness implies the following two properties of η: (1) The measure νη has full topological support in Xη. (2) If Xη is proximal, i.e. if the one-point set {…000…} is contained in Xη and is the unique minimal subset of Xη, then Xη is hereditary, i.e. if x∈Xη and if w is an arbitrary element of {0,1}ℤ, then also the coordinate-wise product w⋅x belongs to Xη. This strengthens two results from [Bartnicka et al. 2015] which need the stronger assumption that has light tails for the same conclusions.
APA:
Keller, G. (2019). Tautness of sets of multiples and applications to B-free dynamics. Studia Mathematica, 247, 205-216. https://dx.doi.org/10.4064/sm180305-9-4
MLA:
Keller, Gerhard. "Tautness of sets of multiples and applications to B-free dynamics." Studia Mathematica 247 (2019): 205-216.
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