Fuglede's spectral set conjecture for convex polytopes
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 $\mathbb{R}^3$.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 DOI:
 10.48550/arXiv.1602.08854
 arXiv:
 arXiv:1602.08854
 Bibcode:
 2016arXiv160208854G
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Functional Analysis
 EPrint:
 To appear in Analysis &