Fuglede's conjecture is false in 5 and higher dimensions

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 :

[1]  N. Katz,et al.  Convex bodies with a point of curvature do not have Fourier bases , 1999, math/9911167.

[2]  Orthogonal harmonic analysis of fractal measures , 1998 .

[3]  Yang Wang,et al.  Universal spectra, universal tiling sets and the spectral set conjecture , 2001 .

[4]  Jeffrey C. Lagarias,et al.  Keller’s cube-tiling conjecture is false in high dimensions , 1992 .

[5]  Tiling and spectral properties of near-cubic domains , 2001, math/0106012.

[6]  I. Laba.  The spectral set conjecture and multiplicative properties of roots of polynomials , 2000, math/0010169.

[7]  Sergei Konyagin,et al.  Institute for Mathematical Physics Spectra of Certain Types of Polynomials and Tiling of Integers with Translates of Finite Sets Spectra of Certain Types of Polynomials and Tiling of Integers with Translates of Finite Sets , 2022 .

[8]  Bent Fuglede,et al.  Commuting self-adjoint partial differential operators and a group theoretic problem , 1974 .

[9]  Jeffrey C. Lagarias,et al.  Spectral Sets and Factorizations of Finite Abelian Groups , 1997 .

[10]  M. Kolountzakis,et al.  Packing, Tiling, Orthogonality and Completeness , 1999, math/9904066.

[11]  Yang Wang,et al.  Wavelets, tiling, and spectral sets , 2002 .

[12]  I. Łaba.  The Spectral Set Conjecture and Multiplicative Properties of Roots of Polynomials , 2002 .

[13]  Non-symmetric convex domains have no basis of exponentials , 1999, math/9904064.

[14]  Fuglede conjecture holds for convex planar domains , 2001, math/0104087.

[15]  Mihail N. Kolountzakis.  The Study of Translational Tiling with Fourier Analysis , 2004 .

[16]  Spectral and Tiling properties of the Unit Cube , 2001, math/0104093.

[17]  J. Reeds,et al.  Orthonormal bases of exponentials for the n-cube , 2000 .

[18]  I. Laba,et al.  Fuglede’s conjecture for a union of two intervals , 2000, math/0002067.

[19]  J. Lagarias,et al.  Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups , 2000, math/0008132.

[20]  A class of non-convex polytopes that admit no orthonormal basis of exponentials , 2001, math/0101217.