Characterizations of democratic systems of translates on locally compact abelian groups

TL;DR

This paper characterizes the democratic property of systems of translates on locally compact abelian groups, generalizing previous results and exploring specific lattice structures through examples and counterexamples.

Contribution

It provides a generalization of democratic system characterizations from integer translates to broader locally compact abelian groups, including torsion lattices.

Findings

Generalized democratic property characterizations to locally compact abelian groups

Identified limitations and possibilities for characterizations in torsion lattice structures

Provided examples and counterexamples illustrating the theoretical results

Abstract

We present characterizations of democratic property for systems of translates on a general locally compact abelian group, along a lattice in that group. That way we generalize the results from [11] on systems of integer translates. Furthermore, we investigate the possibilities of more operative characterizations for lattices with torsion group structure, mainly through examples and counterexamples.