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Fourier quasicrystals and Lagarias' conjecture


Author: S. Yu. Favorov
Journal: Proc. Amer. Math. Soc. 144 (2016), 3527-3536
MSC (2010): Primary 52C23, 42B10, 43A60, 42A75
DOI: https://doi.org/10.1090/proc/12979
Published electronically: February 2, 2016
MathSciNet review: 3503720
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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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Additional Information

S. Yu. Favorov
Affiliation: Karazin’s Kharkiv National University, Svobody sq., 4, 61022, Kharkiv, Ukraine
Email: sfavorov@gmail.com

DOI: https://doi.org/10.1090/proc/12979
Keywords: Quasicrystal, Lagarias' conjecture, unbounded measure, temperate distribution, Fourier transform of measure, almost periodic measure, full-rank lattice
Received by editor(s): April 5, 2015
Received by editor(s) in revised form: August 14, 2015, and September 29, 2015
Published electronically: February 2, 2016
Communicated by: Yingfei Yi
Article copyright: © Copyright 2016 American Mathematical Society