Growth of masses of crystalline measures
Studia Mathematica
MSC: Primary 42A38; Secondary 42A75, 52C23
DOI: 10.4064/sm250325-8-1
Published online: 27 July 2026
Abstract
Let $\mu $ be a measure on the Euclidean space $\mathbb {R}^d$ of unbounded total variation that is positive or translation bounded, and suppose its Fourier transform $\hat \mu $ in the sense of distributions is a pure point measure. 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.
