Tiles with no spectra

Mihail N. Kolountzakis, M. Matolcsi

Research output: Contribution to journalArticle

71 Citations (Scopus)

Abstract

We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their L2 space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang [7]. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. We extend this last example to the groups ℤd and ℝd (for d ≥ 5) thus disproving one direction of the Spectral Set Conjecture of Fuglede [1]. The other direction was recently disproved by Tao [12].

Original languageEnglish
Pages (from-to)519-528
Number of pages10
JournalForum Mathematicum
Volume18
Issue number3
DOIs
Publication statusPublished - Jan 5 2006

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Tile
Finite Abelian Groups
Spectral Set
Orthogonal Basis
Disprove
Tiling
Complement
Subset

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Tiles with no spectra. / Kolountzakis, Mihail N.; Matolcsi, M.

In: Forum Mathematicum, Vol. 18, No. 3, 05.01.2006, p. 519-528.

Research output: Contribution to journalArticle

Kolountzakis, Mihail N. ; Matolcsi, M. / Tiles with no spectra. In: Forum Mathematicum. 2006 ; Vol. 18, No. 3. pp. 519-528.
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