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Let $S(z)$ be an absolutely convergent Dirichlet series with a boundedspectrum and only real zeros $a_n$, let $\mu$ be the sum of unit masses atpoints $a_n$. It is proven that the Fourier transform $\hat\mu$ in the sense ofdistributions is a purely point measure. Conversely, in terms of the propertiesof $\hat\mu$, a sufficient condition was found when $a_n$ are zeros of anabsolutely convergent Dirichlet series with bounded spectrum. Also, in terms ofthe properties of $\hat\mu$, a criterion is established that $a_n$ are zeros ofan almost periodic entire function of exponential growth. In all cases, themultiplicity of zeros is taken into account.  Almost periodic sets, introduced by B.Levin and M.Krein in 1948, playimportant role in our investigations. In particular, we show a new simplerepresentation of such sets.

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