  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.
