Gap distributions of Fourier quasicrystals via LeeYang polynomials
Abstract
Recent work of Kurasov and Sarnak provides a method for constructing onedimensional Fourier quasicrystals (FQ) from the torus zero sets of a special class of multivariate polynomials called LeeYang polynomials. In particular, they provided a nonperiodic FQ with unit coefficients and uniformly discrete support, answering an open question posed by Meyer. Their method was later shown to generate all onedimensional Fourier quasicrystals with $\mathbb{N}$valued coefficients ($ \mathbb{N} $FQ). In this paper, we characterize which LeeYang polynomials give rise to nonperiodic $ \mathbb{N} $FQs with unit coefficients and uniformly discrete support, and show that this property is generic among LeeYang polynomials. We also show that the infinite sequence of gaps between consecutive atoms of any $\mathbb{N}$FQ has a welldefined distribution, which, under mild conditions, is absolutely continuous. This generalizes previously known results for the spectra of quantum graphs to arbitrary $\mathbb{N}$FQs. Two extreme examples are presented: first, a sequence of $\mathbb{N}$FQs whose gap distributions converge to a Poisson distribution. Second, a sequence of random LeeYang polynomials that results in random $\mathbb{N}$FQs whose empirical gap distributions converge to that of a random unitary matrix (CUE).
 Publication:

arXiv eprints
 Pub Date:
 July 2023
 DOI:
 10.48550/arXiv.2307.13498
 arXiv:
 arXiv:2307.13498
 Bibcode:
 2023arXiv230713498A
 Keywords:

 Mathematical Physics;
 Mathematics  Dynamical Systems;
 Mathematics  Number Theory;
 42B10;
 52C23;
 34B45