Fuglede's conjecture is false in 5 and higher dimensions

2003/06/08 by Terence Tao, Tao, Terence · 7 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #Mathematical Analysis and Transform Methods #math.CA #math.CO #msc:20K01 #msc:42B99

paper · pdf · doi:10.48550/arxiv.math/0306134

8 pages, no figures, 2 matrices; submitted, Math Research Letters

arxiv created 2003/06/09 · arxiv updated 2009/11/30

Abstract

We give an example of a set Ω⊂ \R⁵ which is a finite union of unit cubes, such that L²(Ω) admits an orthonormal basis of exponentials \\frac1|Ω|^1/2 e^{2πi ξ_j ⋅ x}: ξ_j ∈ Λ\ for some discrete set Λ⊂ \R⁵, but which does not tile \R⁵ by translations. This answers a conjecture of Fuglede in the negative, at least in 5 and higher dimensions.

Citations

• Fuglede conjecture holds for convex planar domains

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• Some Plancherel identities for unbounded subsets of \mathbb R in duality
• Spectrality of Prime Size Tiles
• On the Spectral Properties of a Class of Planar Sierpinski Self-Affine Measures
• Spectrality of Product Sets with a Perturbed Interval Factor

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