Spectral sets and weak tiling

TL;DR

This paper explores the properties of weak tiling, a geometric condition related to spectral sets, and applies it to various domains including convex bodies, polytopes, and Cantor sets, advancing understanding of Fuglede's conjecture.

Contribution

It extends the study of weak tiling properties and demonstrates their applications across different classes of sets, including non-convex and fractal domains.

Findings

Weak tiling provides a geometric necessary condition for spectrality.

Applications to convex bodies support Fuglede's conjecture in specific cases.

Extensions to non-convex polytopes and Cantor sets reveal broader relevance.

Abstract

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that $\Omega$ is spectral if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it was recently proved that the Fuglede conjecture does hold for the class of convex bodies in $\mathbb{R}^d$. The proof was based on a new geometric necessary condition for spectrality, called "weak tiling". In this paper we study further properties of the weak tiling notion, and present applications to convex bodies, non-convex polytopes, product domains and Cantor sets of positive measure.