Tautness of sets of multiples and applications to B-free dynamics

Beitrag in einer Fachzeitschrift
(Online-Publikation)


Details zur Publikation

Autorinnen und Autoren: Keller G
Zeitschrift: Studia Mathematica
Jahr der Veröffentlichung: 2019
Band: 247
Seitenbereich: 205-216
ISSN: 0039-3223
Sprache: Englisch


Abstract

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.






FAU-Autorinnen und Autoren / FAU-Herausgeberinnen und Herausgeber

Keller, Gerhard Prof. Dr.
Professur für Mathematik (Ergodentheorie)


Zitierweisen

APA:
Keller, G. (2019). Tautness of sets of multiples and applications to B-free dynamics. Studia Mathematica, 247, 205-216.

MLA:
Keller, Gerhard. "Tautness of sets of multiples and applications to B-free dynamics." Studia Mathematica 247 (2019): 205-216.

BibTeX: 

Zuletzt aktualisiert 2019-22-05 um 13:38