Irregular $\mathcal{B}$-free Toeplitz sequences via Besicovitch's construction of sets of multiples without density

TL;DR

This paper constructs specific sets of integers with no asymptotic density that generate irregular and Toeplitz sequences with diverse ergodic properties, advancing understanding of their dynamical and measure-theoretic behavior.

Contribution

It introduces new constructions of sets of multiples without density that produce Toeplitz sequences with various ergodic and spectral properties.

Findings

Constructed sets yield Toeplitz sequences with positive entropy measures.

Produced irregular Toeplitz sequences with uniquely ergodic orbit closures.

Demonstrated sequences with both discrete and positive entropy spectral measures.

Abstract

Modifying Besicovitch's construction of a set $\mathcal{B}$ of positive integers whose set of multiples $\mathcal{M}_{\mathcal{B}}$ has no asymptotic density, we provide examples of such sets $\mathcal{B}$ for which $\eta:=1_{\mathbb{Z}\setminus\mathcal{M}_{\mathcal{B}}}\in\{0,1\}^{\mathbb{Z}}$ is a Toeplitz sequence. Moreover our construction produces examples, for which $\eta$ 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 Lema\'nczyk, we construct a set $\mathcal{B}$ for which $\eta$ is an irregular Toeplitz sequence but for which the orbit closure of $\eta$ in $\{0,1\}^{\mathbb{Z}}$ is uniquely ergodic.