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 $\Omega \subset \mathbb {R}^4$ such that $L^2(\Omega )$ admits an orthonormal basis of exponentials $\{\frac {1}{|\Omega |^{1/2}}e^{2\pi i \langle x, \xi \rangle }\}_{\xi \in \Lambda }$ for some set $\Lambda \subset \mathbb {R}^4$, but which does not tile $\mathbb {R}^4$ 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.