Growth of masses of crystalline measures

Abstract

Let $\mu$ be a measure on the Euclidean space ${\mathbb{R}}^d$ of unbounded total variation that is positive or translation bounded and has a pure point Fourier transform in the sense of distributions $\hat{\mu }$. We prove that the measure $\nu$ with the same support as $\hat{\mu }$ and masses equal to the squares of the masses of $\hat{\mu }$ is translation bounded. We also prove that if $\mu$ is as above and the restriction of its spectrum, i.e., of the support of $\hat{\mu }$, to each ball of fixed radius is a linearly independent set over $\mathbb{Z}$, then the measure $\hat{\mu }$ is also translation bounded. These results imply certain conditions for a crystalline measure to be a Fourier quasicrystal.

9 pages, 21 references