Analogues of Fourier quasicrystals for a strip

We study a certain family of discrete measures with unit masses on a horizontal strip as an analogue of Fourier quasicrystals on the real line. We prove a one-to-one correspondence between supports of measures from this family and zero sets of exponential polynomials with imaginary frequencies. This result is the special case of a general result on measures whose supports correspond to zero sets of absolutely convergent Dirichlet series with bounded spectrum.