Paper

Fuglede's spectral set conjecture for convex polytopes

Lev, Nir, Greenfeld, Rachel

2017

Overview

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\\).

• Publisher: Cornell University Library, arXiv.org
• Subject: Exponential functions / Polytopes / Spectra / Tiles / Translations