Almost periodic distributions and crystalline measures

We study temperate distributions and measures with discrete support in Euclidean space and their Fourier transformswith special attention to almost periodic distributions. In particular, we prove that if distances between points of the support of a measure do not quickly approach 0 at infinity, then this measure is a Fourier quasicrystal (Theorem 1). We also introduce a new class of almost periodicity of distributions,close to the previous one, and study its properties.Actually, we introduce the concept of s-almost periodicity of temperate distributions. We establish the conditions for a measure $\mu$ to be s-almost periodic (Theorem 2), a connection between s-almost periodicityand usual almost periodicity of distributions (Theorem 3). We also prove that the Fourier transform of an almost periodic distribution with locally finite support is a measure (Theorem 4),and prove a necessary and sufficient condition on a locally finite set $E$ for each measure with support on $E$ to have s-almost periodic Fourier transform (Theorem 5).