Fuglede's conjecture is false in 5 and higher dimensions

Abstract

We give an example of a set Ω R5 which is a finite union of unit cubes, such that L2(Ω) admits an orthonormal basis of exponentials {1/|Ω|1/2e2πiξj·x : ξj ∈ Λ} for some discrete set Λ ⊂ R 5, but which does not tile R5 by translations. This answers (one direction of) a conjecture of Fuglede [1] in the negative, at least in 5 and higher dimensions.