Almost Periodic Measures and Meyer Sets

TL;DR

This paper develops a framework connecting almost periodic measures, Meyer sets, and cut and project schemes, providing new characterizations of these structures in locally compact Abelian groups.

Contribution

It introduces a novel construction of cut and project schemes from families of sets and characterizes weighted Dirac combs and Meyer sets using almost periodicity.

Findings

Characterization of weighted Dirac combs via almost periodicity.

Construction of cut and project schemes from set families.

Characterization of Meyer sets in locally compact Abelian groups.

Abstract

In the first part, we construct a cut and project scheme from a family $\{P_\varepsilon\}$ of sets verifying four conditions. We use this construction to characterize weighted Dirac combs defined by cut and project schemes and by continuous functions on the internal groups in terms of almost periodicity. We are also able to characterise those weighted Dirac combs for which the internal function is compactly supported. Lastly, using the same cut and project construction for $\varepsilon$-dual sets, we are able to characterise Meyer sets in $\sigma$-compact locally compact Abelian groups.