Vol. 10, No. 6, 2017

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Fuglede's spectral set conjecture for convex polytopes

Rachel Greenfeld and Nir Lev

Vol. 10 (2017), No. 6, 1497–1538
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 .

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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