Fourier quasicrystals and distributions on Euclidean spaces with spectrum of bounded density

TL;DR

This paper investigates the structure of temperate distributions with discrete support and spectrum of bounded density, establishing conditions under which they decompose into derivatives of lattice Dirac combs, using properties of almost periodic distributions.

Contribution

It provides new conditions characterizing when such distributions can be expressed as finite sums of derivatives of lattice Dirac combs, advancing understanding of their structure.

Findings

Distributions with discrete support and spectrum can be decomposed into derivatives of lattice Dirac combs.

Conditions on coefficients determine the finite sum representation.

Properties of almost periodic distributions are key to these results.

Abstract

We consider temperate distributions on Euclidean spaces with uniformly discrete support and locally finite spectrum. We find conditions on coefficients of distributions under which they are finite sum of derivatives of generalized lattice Dirac combs. These theorems are derived from properties of families of discretely supported measures and almost periodic distributions.