Application of methods of quasicrystals theory to entire functions of exponential growth

Let f be an entire almost periodic function with zeros in a horizontal strip of finite width; for example, any exponential polynomial with purely imaginary exponents is such a function. Let μ be the measure on the set of zeros of f whose masses coincide with multiplicities of zeros. We define the Fourier transform in the sense of distributions for μ and prove that it is a pure point measure on whose complex masses correspond to coefficients of Dirichlet series of the logarithmic derivative of f. Bases on this description and Meyer's theorem on quasicrystals, we give a simple necessary and sufficient condition for f to be a finite product of sines.