Spectrality of product domains and Fuglede's conjecture for convex polytopes

TL;DR

This paper proves that in two dimensions, a convex polygon is spectral if and only if it can tile space by translations, extending previous results and connecting spectrality with tiling properties of convex polytopes.

Contribution

It establishes the spectrality-tile equivalence for convex polygons in two dimensions, confirming the conjecture for this class of convex polytopes.

Findings

Convex polygons in 2D are spectral if and only if they tile space by translations.

Supports the conjecture linking spectral sets and tiling for convex polytopes.

Extends previous results from intervals to convex polygons in two dimensions.

Abstract

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set $\Omega = A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope $\Omega$ is spectral if and only if it can tile by translations.