Fuglede's conjecture is false in 5 and higher dimensions
Abstract
We give an example of a set $\Omega \subset \R^5$ 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 $\Lambda \subset \R^5$, but which does not tile $\R^5$ by translations. This answers a conjecture of Fuglede in the negative, at least in 5 and higher dimensions.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2003
 arXiv:
 arXiv:math/0306134
 Bibcode:
 2003math......6134T
 Keywords:

 Combinatorics;
 Classical Analysis and ODEs;
 20K01;
 42B99
 EPrint:
 8 pages, no figures, 2 matrices