Some properties of Fourier quasicrystals and measures on a strip

In our paper we extend some results of the theory of Fourier quasicrystals on the real line to a horizontal strip of finite width. For measures in a strip we use a natural generalization of the usual Fourier transform for measures on the line. We consider positive or translation bounded measures $\mu$ on a strip whose Fourier transform is a pure point measure $\widehat{\mu }={\sum }_{\gamma \in \mathrm{\Gamma }}{b}_{\gamma }{\delta }_{\gamma }$ (as usual, ${\delta }_{\gamma }$ is the unit mass at the point $\gamma$). We prove that the measure $\nu ={\sum }_{\gamma \in \mathrm{\Gamma }}\mathrm{\mid }{b}_{\gamma }{\mathrm{\mid }}^2{\delta }_{\gamma }$ has the exponential growth. Moreover, if for some $\eta >0$ the points of $\mathrm{\Gamma }$ in every interval of length $\eta$ are linearly independent over integers, then the measure $\widehat{\mu }$ also has the exponential growth.