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arXiv · 2503.19567

Growth of masses of crystalline measures

Peter Boyvalenkov·Sergii Yu. Favorov
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Abstract

Let $\mu$ be a measure on the Euclidean space $\R^d$ 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 $\Z$, then the measure $\hat\mu$ is also translation bounded. These results imply certain conditions for a crystalline measure to be a Fourier quasicrystal.

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Peter Boyvalenkov, Sergii Yu. Favorov. 2025-03-25. Growth of masses of crystalline measures. https://arxiv.org/abs/2503.19567

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