Fuglede's conjecture fails in dimension 4

Research output: Contribution to journalArticle

47 Citations (Scopus)

Abstract

In this note we modify a recent example of Tao and give an example of a set Ω ⊂ ℝ4 such that L2(Ω) admits an orthonormal basis of exponentials {1/|Ω|1/2e 2πi〈x,ξ〉}ξ∈Λ for some set Λ ⊂ ℝ4, but which does not tile ℝ4 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.

Original languageEnglish
Pages (from-to)3021-3026
Number of pages6
JournalProceedings of the American Mathematical Society
Volume133
Issue number10
DOIs
Publication statusPublished - Oct 2005

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Tile
Spectral Set
Orthonormal basis

Keywords

  • Fuglede's conjecture
  • Hadamard matrices
  • Spectral sets
  • Translational tiles

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Fuglede's conjecture fails in dimension 4. / Matolcsi, M.

In: Proceedings of the American Mathematical Society, Vol. 133, No. 10, 10.2005, p. 3021-3026.

Research output: Contribution to journalArticle

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