Fuglede’s conjecture fails in dimension 4

Abstract

In this note we modify a recent example of Tao and give an example of a set Ω ⊂ R 4 \Omega \subset \mathbb {R}^4 such that L 2 ( Ω ) L^2(\Omega ) admits an orthonormal basis of exponentials { 1 | Ω | 1 / 2 e 2 π i ⟨ x , ξ ⟩ } ξ ∈ Λ \{\frac {1}{|\Omega |^{1/2}}e^{2\pi i \langle x, \xi \rangle }\}_{\xi \in \Lambda } for some set Λ ⊂ R 4 \Lambda \subset \mathbb {R}^4 , but which does not tile R 4 \mathbb {R}^4 by translations. This shows that one direction of Fuglede’s conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved.