<p>We study a certain family of discrete measures with unit masseson a horizontal strip as an analogue of Fourier quasicrystals on the real line. Weprove a one-to-one correspondence between supports of measures from this familyand zero sets of exponential polynomials with imaginary frequencies. This resultis the special case of a general result on measures whose supports correspond tozero sets of absolutely convergent Dirichlet series with bounded spectrum.</p>