UNIVERSITY OF CRETE - DEPARTMENT OF MATHEMATICS
Knossos Ave., GR 714 09 Iraklio, Crete, Greece, Tel: +30 2810393800, Fax: +30 2810393881


ANALYSIS SEMINAR

Talk by Maté Matolcsi

THE CONJECTURE OF FUGLEDE FAILS IN DIMENSION FOUR

5 May 2004

In this talk 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 $\{\frac{1}{\vert\Omega \vert^{1/2}}e^{2\pi i \langle x, \xi \rangle }\}_{\xi\in\L }$ for some set $\L\subset{\mathbb{R}}^4$, 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.


Mihalis Kolountzakis 2004-04-30