Tiling by lattices for locally compact abelian groups

TL;DR

This paper provides a straightforward harmonic analysis proof that a bounded domain tiles a locally compact abelian group via a lattice if and only if its dual lattice characters form an orthogonal basis of the domain's L^2 space.

Contribution

It offers a simple, harmonic analysis-based proof of the tiling characterization for bounded domains in locally compact abelian groups.

Findings

Tiling by lattices is characterized by dual lattice characters forming an orthogonal basis.

The proof simplifies existing arguments using harmonic analysis techniques.

The result applies to general locally compact abelian groups.

Abstract

For a locally compact abelian group $G$ a simple proof is given for the known fact that a bounded domain $\Omega$ tiles $G$ with translations by a lattice $\Lambda$ if and only if the set of characters of $G$ indexed by the dual lattice of $\Lambda$ is an orthogonal basis of $L^2(\Omega).$ The proof uses simple techniques from Harmonic Analysis.