Functions tiling with several lattices

TL;DR

This paper investigates the problem of constructing functions with minimal support that tile Euclidean spaces with multiple lattices, providing bounds and exploring finite group analogs.

Contribution

It introduces new bounds and constructions for functions tiling with multiple lattices, and extends the problem to finite abelian groups.

Findings

Established upper bounds via constructions

Derived lower bounds on support size

Explored finite abelian group setting

Abstract

We study the problem of finding a function $f$ with ``small support'' that simultaneously tiles with finitely many lattices $\Lambda_1, \ldots, \Lambda_N$ in $d$-dimensional Euclidean spaces. We prove several results, both upper bounds (constructions) and lower bounds on how large this support can and must be. We also study the problem in the setting of finite abelian groups, which turns out to be the most concrete setting. Several open questions are posed.