Generalized Fourier quasicrystals, almost periodic sets, and zeros of Dirichlet series

Let $S(z)$ be an absolutely convergent Dirichlet series with a bounded spectrum and only real zeros $a_n$, let $\mu$ be the sum of unit masses at points $a_n$. It is proven that the Fourier transform $\widehat{\mu }$ in the sense of distributions is a purely point measure. Conversely, in terms of the properties of $\widehat{\mu }$, a sufficient condition was found when $a_n$ are zeros of an absolutely convergent Dirichlet series with bounded spectrum. Also, in terms of the properties of $\widehat{\mu }$, a criterion is established that $a_n$ are zeros of an almost periodic entire function of exponential growth. In all cases, the multiplicity of zeros is taken into account. Almost periodic sets, introduced by B.Levin and M.Krein in 1948, play important role in our investigations. In particular, we show a new simple representation of such sets.