Temperate distributions with locally finite support and spectrum on Euclidean spaces

TL;DR

This paper proves that certain temperate distributions with discrete support and spectrum in Euclidean spaces are composed of finite unions of lattice translations, extending results known for Fourier quasicrystals using almost periodic distribution techniques.

Contribution

It generalizes the theorem for Fourier quasicrystals to a broader class of temperate distributions with discrete support and spectrum.

Findings

Supports are finite unions of lattice translations.

Extends Fourier quasicrystal results to more general distributions.

Uses almost periodic distribution techniques.

Abstract

We prove that supports of a wide class of temperate distributions with uniformly discrete support and spectrum on Euclidean spaces are finite unions of translations of full-rank lattices. This result is a generalization of the corresponding theorem for Fourier quasicrystals, and its proof uses the technique of almost periodic distributions.