Fuglede's conjecture is false in 5 and higher dimensions

By Terence Tao. 2003

437 citations recorded by the source corpus.

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 2πiξj ·x :

Subfields: Algebra and Number Theory, Applied Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/1b6aa2535d1b6288d2f1add5af821c85561d39b9

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In Algebra and Number Theory

In Applied Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

Fuglede's conjecture on cyclic groups of order pnq (2016, background): “Against the current of research on this subject, Tao disproved this conjecture for dimensions 5 and above [15], constructing a spectral set in 5 dimensions that does not tile the space1.”
Exponential spectra in L2(μ) (2013, background): “One of the interesting and basic questions in non-harmonic Fourier analysis is: What kind of compactly supported probability measures in Rd belong to the above classes of measures? When μ is the restriction of the Lebesgue measure on K with positive measure, the question whether it is a spectral mea”
Analysis of orthogonality and of orbits in affine iterated function systems (2006, methodology): “Since then they have been used extensively in harmonic analysis, in varied contexts involving the analysis of Fourier bases and geometry in for example the papers [Haa96, IoPe98, IoKaPe99, PeWa01, LaWa96, LaWa97, Tao04] among others.”
Spectra of Cantor measures (2014, background): “The spectral set conjecture has been proved to be false by Tao and others in dimension 3 or higher [18,29], while it re mains open in dimensions one and two, see for instance [15, 16, 20, 21, 28] and referenc s therein.”
Combining Riesz bases (2012, neutral): “See also [6,8,14,27,28].”
ON THE SPECTRA OF A CANTOR MEASURE (2008, background): “This conjecture was proved to be false in higher dimensions by Tao [22] and others, but it is still open in dimension 1 and 2.”
The Fuglede conjecture for convex domains is true in all dimensions (2019, background): “ults establishing connections between spectrality and tiling had since been obtained. Nowadays we know, however, that the conjecture is not true without imposing extra assumptions on the domain Ω. In [Tao04], Tao disproved the “spectral implies tiling” part of the conjecture in dimensions 5 and high”
A Concise Guide to Complex Hadamard Matrices (2005, background): “This very problem was intensively studied by high energy physicists investigating the parity violation and analyzing the Cabibbo–Kobayashi–Maskawa matrices [25, 26, 27, 28].”
FOURIER SERIES ON FRACTALS: A PARALLEL WITH WAVELET THEORY (2007, methodology): “For some of these examples we refer the reader to [FMM06, Tao04, LW95, LW96b, LW96c, LW96a, IKT03, Mat05, KL96].”
Construction, classification and parametrization of complex Hadamard matrices (2011, methodology): “A rather surprising application of complex Hadamard matrices has been found recently by Tao [140].”
Defect of a unitary matrix (2007, neutral): “trix’ S 6 : S 6 = 1 √ 6 · 1 1 1 1 1 1 1 1 ω ω ω 2ω 1 ω 1 ω2 ω2 ω 1 ω ω2 1 ω ω2 1 ω 2ω ω 1 ω 1 ω 2ω ω ω 1 , where ω = exp i· 2π 3 , (128) found independently by Tao [32] and by Moorhouse [31] (denoted by S(0) 6 in our catalogue [15]), has the defect equal to zero, so Theorem 4.11 ca”