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 ${\mathbb{R}}^3$.

To appear in Analysis & PDE

References in corpus (2)

• Spectrality and tiling by cylindric domains
• Tilings by translation

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• The spectrality of self-affine measure under the similarity transformation of $G{L}_n(p)$
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• Multiplicity of nontrivial zeros of primitive L-functions via higher-level correlations
• Fuglede's conjecture holds in ${\mathbb{Z}}_p\times {\mathbb{Z}}_{p^n}$