Fuglede's spectral set conjecture for convex polytopes

TL;DR

This paper proves Fuglede's spectral set conjecture for convex polytopes in three-dimensional space, showing that spectral convex polytopes must tile space by translations, confirming the conjecture in this case.

Contribution

The authors establish that in three dimensions, spectral convex polytopes necessarily tile space, confirming Fuglede's conjecture for this class of shapes.

Findings

Spectral convex polytopes in 3D tile space by translations.

All facets of spectral convex polytopes are centrally symmetric.

Fuglede's conjecture holds for convex polytopes in d space.

Abstract

Let $\Omega$ be a convex polytope in $\mathbb{R}^d$. We say that $\Omega$ is spectral if the space $L^2(\Omega)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that $\Omega$ is spectral if and only if it can tile the space by translations. It is known that if $\Omega$ tiles then it is spectral, but the converse was proved only in dimension $d=2$, by Iosevich, Katz and Tao. By a result due to Kolountzakis, if a convex polytope $\Omega\subset \mathbb{R}^d$ is spectral, then it must be centrally symmetric. We prove that also all the facets of $\Omega$ are centrally symmetric. These conditions are necessary for $\Omega$ to tile by translations. We also develop an approach which allows us to prove that in dimension $d=3$, any spectral convex polytope $\Omega$ indeed tiles by translations. Thus we obtain that Fuglede's conjecture is true for convex polytopes in $\mathbb{R}^3$.