On Fuglede's conjecture and the existence of universal spectra

Abstract

Recent methods developed by Tao \cite{tao}, Kolountzakis and Matolcsi \cite{nspec} have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in $\RR^5$ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a non-spectral tile. In search of lower dimensional non-spectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang \cite{lagwang}. In particular, we prove here that the USC and the ``tile $\to$ spectral'' direction of Fuglede's conjecture are equivalent in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szabó \cite{lagszab} for the existence of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples of tiles which have no spectra. We overcome these difficulties by invoking some ideas of Révész and Farkas \cite{revfark}, and obtain non-spectral tiles in $\RR^3$. Fuglede's conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The 1 dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz \cite{covmey}.

12 pages

Cited by in corpus (5)

• Fuglede's conjecture holds for cyclic groups of order $pqrs$
• Spectrality of random convolutions generated by finitely many Hadamard triples
• Spectrality of Infinite Convolutions and Random Convolutions
• Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctions
• On product spectral sets and functional tiles in ${\mathbb{Q}}_p^d$