Fuglede’s conjecture is false in 5 and higher dimensions

We give an example of a set $\Omega \subset {\hbox{\bf R}}^{5}$ 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 $\Lambda \subset {\hbox{\bf R}}^{5}$, but which does not tile $\R^{5}$ by translations. This answers (one direction of) a conjecture of Fuglede \cite{fuglede} in the negative, at least in 5 and higher dimensions.

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Published 1 January 2004