Fuglede's conjecture fails in dimension 4

Details Fuglede's conjecture fails in dimension 4 Fuglede's conjecture fails in dimension 4

2006-11-30

arxiv.org/abs/math/0611936v1

Proc. Amer. Math. Soc. 133 (2005), no. 10, 3021--3026

Mathematics

Classical Analysis and ODEs

6 pages

Scientific paper

In this note we give an example of a set $\W\subset \R^4$ such that $L^2(\W)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\W |^{1/2}}e^{2\pi i x, \xi}\}_{\xi\in\L}$ for some set $\L\subset\R^4$, but which does not tile $\R^4$ by translations. This improves Tao's recent 5-dimensional example, and 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.

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