Multidimensional Fourier Quasicrystals I. Sufficient Conditions

TL;DR

This paper establishes sufficient conditions for atomic measures in multidimensional space to be Fourier quasicrystals, extending previous one-dimensional results and utilizing advanced geometric methods involving toric geometry, Grothendieck residues, and Newton polytopes.

Contribution

It generalizes the criteria for Fourier quasicrystals from one dimension to higher dimensions using novel geometric techniques.

Findings

Derived sufficient conditions for multidimensional Fourier quasicrystals.

Extended previous one-dimensional criteria to higher dimensions.

Linked Fourier quasicrystal properties with toric geometry and Newton polytopes.

Abstract

We derive sufficient conditions for an atomic measure $\sum_{\lambda \in \Lambda} m_\lambda\, \delta_\lambda,$ where $\Lambda \subset \mathbb R^n,$ $m_\lambda$ are positive integers, and $\delta_\lambda$ is the point measure at $\lambda,$ to be a Fourier quasicrystal, and suggest why they may also be necessary. These conditions extend the necessary and sufficient conditions derived by Lev, Olevskii, and Ulanovskii for $n = 1.$ Our methods exploit the toric geometry relation between Grothendieck residues and Newton polytopes derived by Gelfond and Khovanskii.