Fuglede's conjecture is false in 5 and higher dimensions

We give an example of a set $\mathrm{\Omega }\subset {\mathbb{R}}^5$ which is a finite union of unit cubes, such that $L^2(\mathrm{\Omega })$ admits an orthonormal basis of exponentials $\{\frac{1}{\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}2}}{e}^{2\pi i{\xi }_j\cdot x}:{\xi }_j\in \mathrm{\Lambda }\}$ for some discrete set $\mathrm{\Lambda }\subset {\mathbb{R}}^5$, 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.