Fuglede's spectral set conjecture for convex polytopes

By Rachel Greenfeld, Nir Lev. 2016

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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 $\mathbb{R}^3$.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/dabf4926ebd4199fe53bf567ba4dfbf3cd0cf361

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

The Fuglede conjecture for convex domains is true in all dimensions (2019, methodology): “e. To the contrary, the “spectral implies tiling” direction of the conjecture for convex bodies was proved only in R2 [IKT03], and also in R3 under the a priori assumption that Ω is a convex polytope [GL17]. In higher dimensions, this direction of the conjecture remained completely open (even in the”