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2017 Fuglede's spectral set conjecture for convex polytopes
Rachel Greenfeld, Nir Lev
Anal. PDE 10(6): 1497-1538 (2017). DOI: 10.2140/apde.2017.10.1497

Abstract

Let Ω be a convex polytope in d . We say that Ω is spectral if the space L2(Ω) admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that Ω is spectral if and only if it can tile the space by translations. It is known that if Ω 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 Ω d is spectral, then it must be centrally symmetric. We prove that also all the facets of Ω are centrally symmetric. These conditions are necessary for Ω to tile by translations.

We also develop an approach which allows us to prove that in dimension d = 3, any spectral convex polytope Ω indeed tiles by translations. Thus we obtain that Fuglede’s conjecture is true for convex polytopes in 3 .

Citation

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Rachel Greenfeld. Nir Lev. "Fuglede's spectral set conjecture for convex polytopes." Anal. PDE 10 (6) 1497 - 1538, 2017. https://doi.org/10.2140/apde.2017.10.1497

Information

Received: 5 March 2017; Accepted: 29 May 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1377.42014
MathSciNet: MR3678495
Digital Object Identifier: 10.2140/apde.2017.10.1497

Subjects:
Primary: 42B10 , 52C22

Keywords: convex polytope , Fuglede's conjecture , spectral set , tiling

Rights: Copyright © 2017 Mathematical Sciences Publishers

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