The DOI matches its Crossref record; 19 of the 21 references it cites were found in a registry or an index.
Fuglede’s conjecture is false in 5 and higher dimensions
Open accessAbstract
We give an example of a set Ω ⊂ R 5 which is a finite union of unit cubes, such that L 2 (Ω) admits an orthonormal basis of exponentials { 1 |Ω| 1/2 e 2πiξ j •x : ξ j ∈ Λ} for some discrete set Λ ⊂ R 5 , but which does not tile R 5 by translations.This answers (one direction of) a conjecture of Fuglede [1] in the negative, at least in 5 and higher dimensions.
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Cites
Commuting self-adjoint partial differential operators and a group theoretic problem
Fuglede, B. (1974) · Journal of Functional Analysis · doi:10.1016/0022-1236(74)90072-x · Cited by 543
Fuglede’s conjecture for a union of two intervals
Łaba, I. (2001) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-01-06035-x · Cited by 141
The Spectral Set Conjecture and Multiplicative Properties of Roots of Polynomials
Łaba, I. (2002) · Journal of the London Mathematical Society · doi:10.1112/s0024610702003149 · Cited by 122
Keller’s cube-tiling conjecture is false in high dimensions
Lagarias, J. C., & Shor, P. W. (1992) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-1992-00318-x · Cited by 114
Spectral Sets and Factorizations of Finite Abelian Groups
Lagarias, J. C., & Wang, Y. (1997) · Journal of Functional Analysis · doi:10.1006/jfan.1996.3008 · Cited by 97
Orthonormal bases of exponentials for the n-cube
Lagarias, J. C., Reeds, J. A., & Wang, Y. (2000) · Duke Mathematical Journal · doi:10.1215/s0012-7094-00-10312-2 · Cited by 96
Convex bodies with a point of curvature do not have Fourier bases
Iosevich, A., Katz, N. H., & Tao, T. (2001) · American Journal of Mathematics · doi:10.1353/ajm.2001.0003 · Cited by 95
Self-citationNon-symmetric convex domains have no basis of exponentials
Kolountzakis, M. N. (2000) · Illinois Journal of Mathematics · doi:10.1215/ijm/1256060414 · Cited by 87
The Study of Translational Tiling with Fourier Analysis
Kolountzakis, M. N. (2004) · Applied and numerical harmonic analysis · doi:10.1007/978-0-8176-8172-2_7 · Cited by 73
Universal spectra, universal tiling sets and the spectral set conjecture
Pedersen, S., & Wang, Y. (2001) · MATHEMATICA SCANDINAVICA · doi:10.7146/math.scand.a-14325 · Cited by 66
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Durt, T., Englert, B., Bengtsson, I., & Życzkowski, K. (2010) · International Journal of Quantum Information · doi:10.1142/s0219749910006502 · Cited by 751
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Kolountzakis, M. N., & Matolcsi, M. (2006) · Forum Mathematicum · doi:10.1515/forum.2006.026 · Cited by 230
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Metadata
- Authors
- Tao, T.
- Affiliations
- University of California, Los Angeles (US)
- Corresponding author
- Tao, Terence
- ORCID
- Tao, Terence
- Venue
- Mathematical Research Letters
- Publisher
- International Press of Boston
- Published
- 2004
- Volume
- 11
- Issue
- 2
- Pages
- 251-258
- ISSN
- 1073-2780 (ISSN-L)
- 1945-001X
- OpenAlex
- W2009600348
- Cited by
- 392
- References
- 21
- Citations by year
- 2026: 17 · 2025: 28 · 2024: 26 · 2023: 33 · 2022: 30 · 2021: 27 · 2020: 22 · 2019: 23 · 2018: 20 · 2017: 18 · 2016: 17 · 2015: 19 · 2014: 11 · 2013: 20 · 2012: 19
- FWCI
- 12.66
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Mathematical Research Letters (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Physical Sciences › Mathematics › Algebra and Number Theory › Analytic Number Theory Research
- Keywords
- Fuglede conjecture, counterexample, unit cube, exponential bases, discrete sets, extra dimensions, L2 spaces
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