  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 $\hat\mu$ in the sense of
distributions is a purely point measure. Conversely, in terms of the properties
of $\hat\mu$, 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 $\hat\mu$, 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.
