Geometric implications of weak tiling

TL;DR

This paper explores the geometric properties of sets that weakly tile their complement, showing convex polytopes must be symmetric with symmetric facets and providing conditions on interval unions in one dimension.

Contribution

It provides new geometric insights into weak tiling, including a direct proof of symmetry for convex polytopes and conditions on interval unions in one dimension.

Findings

Convex polytopes that weakly tile are symmetric with symmetric facets.

Finite unions of intervals in one dimension must satisfy specific gap length conditions.

The paper offers a self-contained proof related to weak tiling properties.

Abstract

The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling and establish some geometric properties of sets that weakly tile their complement. If $A \subset \mathbb{R}^d$ is a convex polytope, we give a direct and self-contained proof that $A$ must be symmetric and have symmetric facets. If $A \subset \mathbb{R}$ is a finite union of intervals, we give a necessary condition on the lengths of the gaps between the intervals.