Bounded common fundamental domains for two lattices

TL;DR

This paper proves the existence of a bounded, measurable common fundamental domain for any two lattices of the same volume in Euclidean space, which can be used to construct orthogonal Gabor bases.

Contribution

It establishes the existence of a finite union of polytopes forming a common fundamental domain for two lattices of equal volume, linking geometric tiling with Gabor analysis.

Findings

Existence of bounded common fundamental domain for two lattices of equal volume.

The fundamental domain can be a finite union of polytopes.

The indicator function of this domain forms a Gabor orthogonal basis.

Abstract

We prove that for any two lattices $L, M \subseteq \mathbb{R}^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E \subseteq \mathbb{R}^d$ such that $E$ tiles $\mathbb{R}^d$ when translated by $L$ or by $M$. In fact, the set $E$ can be taken to be a finite union of polytopes. A consequence of this is that the indicator function of $E$ forms a Weyl--Heisenberg (Gabor) orthogonal basis of $L^2(\mathbb{R}^d)$ when translated by $L$ and modulated by $M^*$, the dual lattice of $M$.