Abstract. We give definitions of crystalline measure and Fourier quasicrystal, discuss
the relationship between these notions, and show some sufficient conditions
for a crystalline measure to be a Fourier quasicrystal. We then present the
theorem of Olevskii and Ulanovskii on the 1-1 correspondence between zeros
of exponential polynomials and Fourier quasicrystals with unit masses. We
show generalizations of this result to the zeros of absolutely convergence
Dirichlet series, to slowly growing measures with countable spectrum, to the
zeros of almost periodic functions in a strip. The last result yields the simple
criterion of representability of almost periodic entire functions of exponential
growth (in particular exponential polynomials) as a finite product of sines.