Fourier quasicrystals with positive integer masses are sections of Lee-Yang cycles
Abstract
We prove that Fourier quasicrystal measures with positive integer masses are precisely the nonzero real sections of effective Lee-Yang cycles by matrices with positive maximal minors, with masses given by weighted local intersection degrees. For Delone support, the reduced support of the cycle can be chosen strict. For unit masses, the cycle can be chosen reduced, with local intersection degree one at every root. Every integer-mass FQ measure has support equal to the common complex zero set of finitely many real-valued trigonometric polynomials. These results answer Questions (1), (2), and (4) of Alon, Kummer, Kurasov, and Vinzant. We recover a finite-dimensional torus from the spectrum and prove that its canonical analytic set is algebraic. A support-height theorem for holonomic sequences provides a linear spectral bound, from which we reconstruct the complex branches and their integer coefficients using fiber moments. Singular sections carry their natural local intersection degrees; discarding these can destroy the Fourier quasicrystal property.
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BibTeX
@misc{wei2026fourier,
title = {{Fourier quasicrystals with positive integer masses are sections of Lee-Yang cycles}},
author = {Dongsheng Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10652},
eprint = {2610.10652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10652},
}
RIS
TY - UNPB
TI - Fourier quasicrystals with positive integer masses are sections of Lee-Yang cycles
AU - Dongsheng Wei
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.10652
UR - https://arxiv.org/abs/2610.10652
AB - We prove that Fourier quasicrystal measures with positive integer masses are precisely the nonzero real sections of effective Lee-Yang cycles by matrices with positive maximal minors, with masses given by weighted local intersection degrees. For Delone support, the reduced support of the cycle can be chosen strict. For unit masses, the cycle can be chosen reduced, with local intersection degree one at every root. Every integer-mass FQ measure has support equal to the common complex zero set of finitely many real-valued trigonometric polynomials. These results answer Questions (1), (2), and (4) of Alon, Kummer, Kurasov, and Vinzant. We recover a finite-dimensional torus from the spectrum and prove that its canonical analytic set is algebraic. A support-height theorem for holonomic sequences provides a linear spectral bound, from which we reconstruct the complex branches and their integer coefficients using fiber moments. Singular sections carry their natural local intersection degrees; discarding these can destroy the Fourier quasicrystal property.
ER -