Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties

TL;DR

This paper constructs higher-dimensional Fourier quasicrystals using Lee-Yang varieties, extending previous one-dimensional work, and demonstrates their properties as Delone almost periodic sets with finite intersections with periodic sets.

Contribution

It introduces a method to generate Fourier quasicrystals in arbitrary dimensions via complex algebraic varieties related to Lee-Yang polynomials, generalizing prior one-dimensional constructions.

Findings

Constructed Fourier quasicrystals in arbitrary dimensions.

Proved these sets are Delone almost periodic.

Showed they have finite intersections with discrete periodic sets.

Abstract

In this paper, we construct Fourier quasicrystals with unit masses in arbitrary dimensions. This generalizes a one-dimensional construction of Kurasov and Sarnak. To do this, we employ a class of complex algebraic varieties avoiding certain regions in $\mathbb{C}^n$, which generalize hypersurfaces defined by Lee-Yang polynomials. We show that these are Delone almost periodic sets that have at most finite intersection with every discrete periodic set.