Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 6 (2017), 1497-1538.

Fuglede's spectral set conjecture for convex polytopes

Rachel Greenfeld and Nir Lev

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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 .

Article information

Source
Anal. PDE, Volume 10, Number 6 (2017), 1497-1538.

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843530

Digital Object Identifier
doi:10.2140/apde.2017.10.1497

Mathematical Reviews number (MathSciNet)
MR3678495

Zentralblatt MATH identifier
1377.42014

Subjects
Primary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 52C22: Tilings in $n$ dimensions [See also 05B45, 51M20]

Keywords
Fuglede's conjecture spectral set tiling convex polytope

Citation

Greenfeld, Rachel; Lev, Nir. Fuglede's spectral set conjecture for convex polytopes. Anal. PDE 10 (2017), no. 6, 1497--1538. doi:10.2140/apde.2017.10.1497. https://projecteuclid.org/euclid.apde/1510843530


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