On Fuglede's conjecture and the existence of universal spectra

Balint Farkas, M. Matolcsi, Peter Mora

Research output: Contribution to journalArticle

40 Citations (Scopus)

Abstract

Recent methods developed by Tao [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in ℝ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an 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 Szabó [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 Révész and Farkas [2], and obtain nonspectral tiles in ℝ. Fuglede's conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The one-dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz [1].

Original languageEnglish
Pages (from-to)483-494
Number of pages12
JournalJournal of Fourier Analysis and Applications
Volume12
Issue number5
DOIs
Publication statusPublished - Oct 2006

Fingerprint

Tile
Spectral Set
Tiling
Counterexample
Necessary
Sufficient Conditions

Keywords

  • Fuglede's conjecture
  • Spectral sets
  • Translational tiles
  • Universal spectrum

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Analysis

Cite this

On Fuglede's conjecture and the existence of universal spectra. / Farkas, Balint; Matolcsi, M.; Mora, Peter.

In: Journal of Fourier Analysis and Applications, Vol. 12, No. 5, 10.2006, p. 483-494.

Research output: Contribution to journalArticle

@article{1e1c1d6d60a845e1b7c7ea11bc95e19f,
title = "On Fuglede's conjecture and the existence of universal spectra",
abstract = "Recent methods developed by Tao [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in ℝ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an 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 Szab{\'o} [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 R{\'e}v{\'e}sz and Farkas [2], and obtain nonspectral tiles in ℝ. Fuglede's conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The one-dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz [1].",
keywords = "Fuglede's conjecture, Spectral sets, Translational tiles, Universal spectrum",
author = "Balint Farkas and M. Matolcsi and Peter Mora",
year = "2006",
month = "10",
doi = "10.1007/s00041-005-5069-7",
language = "English",
volume = "12",
pages = "483--494",
journal = "Journal of Fourier Analysis and Applications",
issn = "1069-5869",
publisher = "Birkhause Boston",
number = "5",

}

TY - JOUR

T1 - On Fuglede's conjecture and the existence of universal spectra

AU - Farkas, Balint

AU - Matolcsi, M.

AU - Mora, Peter

PY - 2006/10

Y1 - 2006/10

N2 - Recent methods developed by Tao [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in ℝ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an 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 Szabó [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 Révész and Farkas [2], and obtain nonspectral tiles in ℝ. Fuglede's conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The one-dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz [1].

AB - Recent methods developed by Tao [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in ℝ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an 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 Szabó [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 Révész and Farkas [2], and obtain nonspectral tiles in ℝ. Fuglede's conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The one-dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz [1].

KW - Fuglede's conjecture

KW - Spectral sets

KW - Translational tiles

KW - Universal spectrum

UR - http://www.scopus.com/inward/record.url?scp=33750501142&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750501142&partnerID=8YFLogxK

U2 - 10.1007/s00041-005-5069-7

DO - 10.1007/s00041-005-5069-7

M3 - Article

VL - 12

SP - 483

EP - 494

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

SN - 1069-5869

IS - 5

ER -