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Fuglede's conjecture fails in dimension 4

Author: Máté Matolcsi
Journal: Proc. Amer. Math. Soc. 133 (2005), 3021-3026
MSC (2000): Primary 42B99; Secondary 20K01
Published electronically: March 24, 2005
MathSciNet review: 2159781
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Abstract: In this note 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 $\{\frac{1}{\vert\Omega \vert^{1/2}}e^{2\pi i \langle x, \xi \rangle }\}_{\xi\in\Lambda}$ for some set $\Lambda\subset\mathbb{R} ^4$, 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.

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Máté Matolcsi
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127 H-1364 Budapest, Hungary

Keywords: Translational tiles, spectral sets, Fuglede's conjecture, Hadamard matrices
Received by editor(s): May 21, 2004
Published electronically: March 24, 2005
Additional Notes: The author was supported by Hungarian Research Funds OTKA-T047276, OTKA-F049457, OTKA-T049301
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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