Fourier quasicrystals and Lagarias’ conjecture
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- by S. Yu. Favorov PDF
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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
- MR Author ID: 189658
- ORCID: 0000-0002-4687-776X
- Email: sfavorov@gmail.com
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3527-3536
- MSC (2010): Primary 52C23, 42B10, 43A60, 42A75
- DOI: https://doi.org/10.1090/proc/12979
- MathSciNet review: 3503720