Fuglede’s spectral set conjecture for convex polytopes

Greenfeld, Rachel; Lev, Nir

doi:10.2140/apde.2017.10.1497

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Fuglede's spectral set conjecture for convex polytopes

Rachel Greenfeld and Nir Lev

Vol. 10 (2017), No. 6, 1497–1538

DOI: 10.2140/apde.2017.10.1497

Abstract

Let $Ω$ be a convex polytope in $ℝ^{d}$ . We say that $Ω$ is spectral if the space $L^{2} (Ω)$ 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 $Ω \subset ℝ^{d}$ 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 $ℝ^{3}$ .

Keywords

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

Mathematical Subject Classification 2010

Primary: 42B10, 52C22

Milestones

Received: 5 March 2017

Accepted: 29 May 2017

Published: 14 July 2017

Authors

Rachel Greenfeld
Department of Mathematics Bar-Ilan University Ramat-Gan 52900 Israel

Nir Lev
Department of Mathematics Bar-Ilan University Ramat-Gan 52900 Israel

Vol. 10, No. 6, 2017