Generalized Fourier quasicrystals and almost periodic sets

Abstract: Let $\mu$ be a positive measure on the real line with locally finite support $\Lambda$ and integer masses such that its Fourier transform in the sense of distributions is a purely point measure. An explicit form is found for an entire almost periodic function with a set of zeros $\Lambda$, taking multiplicities into account. A necessary and sufficient condition for the exponential growth of this function is also found. Our constructions are based on the properties of almost periodic sets on the line. In particular, in the article we find a simple representation of such sets.