Constructing Toeplitz arrays via cut and project schemes

TL;DR

This paper establishes a correspondence between Toeplitz arrays and model sets via cut and project schemes, and constructs examples with specific ergodic measure properties.

Contribution

It introduces a novel connection between Toeplitz arrays and model sets, and constructs irregular Toeplitz arrays with prescribed ergodic measures.

Findings

Established a one-to-one correspondence between Toeplitz arrays and model sets.

Constructed irregular Toeplitz arrays with exactly k ergodic measures.

Produced examples with infinite maximal rank and the same measure-theoretic structure.

Abstract

We show a one-to-one correspondence between Toeplitz arrays over residually finite topological groups and model sets obtained via specific cut and project schemes, built from the odometer associated to the Toeplitz array. As an application, we construct irregular Toeplitz arrays which are extensions of maximal rank $k$ over their maximal equicontinuous factor (given by the associated odometer) and have exactly $k$ different ergodic measures. A modification of the construction also allows to obtain examples with the same measure-theoretic structure, but infinite maximal rank.