Non-negative crystalline and Poisson measures in the Euclidean space

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