Growth of masses of crystalline measures

TL;DR

This work studies measures $\mu$ on ${\mathbb R}^d$ with unbounded total variation that are either positive or translation bounded and possess a pure point Fourier transform $\hat{\mu}$. The authors prove that the spectral masses yield a translation-bounded measure $\nu=\sum_{\gamma\in\Gamma}|b_{\gamma}|^2\delta_{\gamma}$ on the same support $\Gamma$ as $\hat{\mu}$, and they show that if the spectrum is locally linearly independent over ${\mathbb Z}$, then $\hat{\mu}$ is translation bounded as well, providing criteria for crystalline measures to be Fourier quasicrystals. The analysis relies on almost periodic function theory, Parseval-type identities, and a multidimensional Kronecker lemma to connect spectral structure with translation-boundedness. Under additional growth conditions on the spectrum, temperedness of $|\hat{\mu}|$ follows, reinforcing the Fourier quasicrystal structure. Overall, the results establish concrete spectral and translational regularity conditions that guarantee crystalline measures exhibit quasicrystal-like Fourier behavior with potential applications to Poisson-type formulas and spectral synthesis.

Abstract

Let $μ$ 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 $\hatμ$. We prove that the measure $ν$ with the same support as $\hatμ$ and masses equal to the squares of the masses of $\hatμ$ is translation bounded. We also prove that if $μ$ is as above and the restriction of its spectrum, i.e., of the support of $\hatμ$, to each ball of fixed radius is a linearly independent set over $\Z$, then the measure $\hatμ$ is also translation bounded. These results imply certain conditions for a crystalline measure to be a Fourier quasicrystal.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary
• Definition 1
• Proposition 1
• Proposition 2
• Proposition 3