Which Meyer sets are regular model sets? A characterization via almost periodicity

TL;DR

This paper characterizes regular model sets among Meyer sets in locally compact abelian groups by establishing that they are exactly the almost periodic patterns, extending previous Euclidean space results.

Contribution

It proves that in a general setting, Meyer sets are regular model sets if and only if they exhibit almost periodicity, broadening the understanding of their structure.

Findings

Meyer sets in locally compact abelian groups are regular model sets iff they are almost periodic patterns.

The result generalizes previous Euclidean space characterizations to a wider class of groups.

Provides a new criterion for identifying regular model sets based on almost periodicity.

Abstract

In 2012, Meyer introduced the notions of generalized almost periodic measure and almost periodic pattern and proved that regular model sets in Euclidean space are almost periodic patterns. Here, we prove the converse in a slightly more general setting. Specifically, we show that a Meyer set in any $\sigma$-compact locally compact abelian group is a regular model set if and only if it is an almost periodic pattern.