Fourier quasicrystals and Lagarias’ conjecture

Favorov, S.

doi:10.1090/proc/12979

Fourier quasicrystals and Lagarias’ conjecture
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by S. Yu. Favorov PDF

Proc. Amer. Math. Soc. 144 (2016), 3527-3536 Request permission

Abstract:

J. C. Lagarias (2000) conjectured that if $\mu$ is a complex measure on $p$-dimensional Euclidean space with a uniformly discrete support and its Fourier transform in the sense of distributions is also a measure with a uniformly discrete support, then the support of $\mu$ is a subset of a finite union of translates of some full-rank lattice. The conjecture was proved by N. Lev and A. Olevski (2013) in the case $p=1$. In the case of an arbitrary $p$ they proved the conjecture only for a positive measure $\mu$.

Here we show that Lagarias’ conjecture does not always hold and investigate two special types of measure distributions connected with the conjecture.

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