Non-negative crystalline and Poisson measures in the Euclidean space

TL;DR

This paper investigates special non-negative measures in Euclidean space with pure point Fourier transforms, exploring their connection to almost periodicity, uniqueness properties, and conditions linking measures to Dirichlet series zero sets.

Contribution

It introduces new characterizations of non-negative atomic measures with pure point Fourier transforms and links them to almost periodicity and Dirichlet series zero sets.

Findings

Established connection between measures and almost periodicity

Proved several forms of the uniqueness theorem

Derived conditions linking measures to Dirichlet series zero sets

Abstract

We study properties of temperate non-negative purely atomic measures in the Euclidean space such that the distributional Fourier transform of these measures are pure point ones. A connection between these measures and almost periodicity is shown, several forms of the uniqueness theorem are proved. We also obtain necessary and sufficient conditions for a measure with positive integer masses on the real line to correspond the zero set of an absolutely convergent Dirichlet series with bounded spectrum.