A class of non-convex polytopes that admit no orthonormal basis of exponentials

Abstract

A conjecture of Fuglede states that a bounded measurable set D, of measure 1, can tile space by translations if and only if the Hilbert space L^2(D) has an orthonormal basis consisting of exponentials exp(i 2 pi lambda x). If D has the latter property it is called spectral. Let D be a polytope with the following property: there is a direction u such that, of all the polytope faces perpendicular to u, the total area of the faces pointing in the positive u direction is more than the total area of the faces pointing in the negative u direction. It is almost obvious that such a polytope D cannot tile space by translation. We prove in this paper that such a domain is also not spectral, which agrees with Fuglede's conjecture.