Growth of masses of crystalline measures

2025/03/25 by Boyvalenkov, Peter, Favorov, Sergii Yu.
#42A38 #42A75 #52C23 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2503.19567

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

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