### Abstract

In this note we modify a recent example of Tao and give an example of a set Ω ⊂ ℝ^{4} such that L^{2}(Ω) admits an orthonormal basis of exponentials {1/|Ω|^{1/2}e ^{2πi〈x,ξ〉}}_{ξ∈Λ} for some set Λ ⊂ ℝ^{4}, but which does not tile ℝ^{4} 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 language | English |
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Pages (from-to) | 3021-3026 |

Number of pages | 6 |

Journal | Proceedings of the American Mathematical Society |

Volume | 133 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 2005 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 133, no. 10, pp. 3021-3026. https://doi.org/10.1090/S0002-9939-05-07874-3

}

TY - JOUR

T1 - Fuglede's conjecture fails in dimension 4

AU - Matolcsi, M.

PY - 2005/10

Y1 - 2005/10

N2 - 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.

AB - 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.

KW - Fuglede's conjecture

KW - Hadamard matrices

KW - Spectral sets

KW - Translational tiles

UR - http://www.scopus.com/inward/record.url?scp=26444500165&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26444500165&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-05-07874-3

DO - 10.1090/S0002-9939-05-07874-3

M3 - Article

AN - SCOPUS:26444500165

VL - 133

SP - 3021

EP - 3026

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 10

ER -