On Fuglede’s conjecture and the existence of universal spectra

Farkas, Bálint and Matolcsi, Máté and Móra, P. (2006) On Fuglede’s conjecture and the existence of universal spectra. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 12 (5). pp. 483-494. ISSN 1069-5869


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Recent methods developed by, Too [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in R(5) Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed all example of a nonspectral tile. In search of lower dimensional nonspectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang [14]. In particular, we prove here that the USC and the "tile --> spectral " direction of Fuglede's conjecture are equivalent in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szabo [13] 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 Revesz and Farkas [2], and obtain nonspectral tiles in R(3).

Item Type: Article
Uncontrolled Keywords: Universal spectrum; translational tiles; spectral sets; Fuglede's conjecture; TILES; DOMAINS; INTEGERS; DIMENSION-4; Polynomials; NO SPECTRA; SET CONJECTURE; universal spectrum
Subjects: Q Science / természettudomány > QA Mathematics / matematika
Depositing User: MTMT SWORD
Date Deposited: 10 Dec 2013 13:44
Last Modified: 10 Dec 2013 13:44

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