Fuglede's conjecture is false in 5 and higher dimensions

Abstract

We give an example of a set $\Omega \subset {\mathbb{R}}^5$ which is a finite union of unit cubes, such that $L^2(\Omega )$ admits an orthonormal basis of exponentials $\{\frac{1}{|\Omega {|}^{1/2}}{e}^{2\pi i{\xi }_j\cdot x}:{\xi }_j\in \Lambda \}$ for some discrete set $\Lambda \subset {\mathbb{R}}^5$, but which does not tile ${\mathbb{R}}^5$ by translations. This answers a conjecture of Fuglede in the negative, at least in 5 and higher dimensions.

8 pages, no figures, 2 matrices; submitted, Math Research Letters