Irregular B-free Toeplitz sequences via Besicovitch's construction of sets of multiples without density

2021/01/03 by Gerhard Keller, Keller, Gerhard
Mathematics · #37A35 #37A45 #37B05 #37B10 #37B40 (Primary) 11N25 (Secondary) #Advanced Topology and Set Theory #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.DS #msc:11N25 #msc:37A35 #msc:37A45 #msc:37B05 #msc:37B10 #msc:37B40

paper · pdf · doi:10.48550/arxiv.2101.00655

arxiv created 2021/01/03 · openalex publication_date 2021/01/03 · arxiv updated 2021/01/05 · openalex created_date 2022/09/04 · openalex updated_date 2026/07/28

Abstract

Modifying Besicovitch's construction of a set B of positive integers whose set of multiples M_B has no asymptotic density, we provide examples of such sets B for which η:=1_ℤ\setminusM_B∈\0,1\^ℤ is a Toeplitz sequence. Moreover our construction produces examples, for which η is not only quasi-generic for the Mirsky measure (which has discrete dynamical spectrum), but also for some measure of positive entropy. On the other hand, modifying slightly an example from Kasjan, Keller, and Lemańczyk, we construct a set B for which η is an irregular Toeplitz sequence but for which the orbit closure of η in \0,1\^ℤ is uniquely ergodic.

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