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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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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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