Fuglede’s conjecture fails in dimension 4

By Máté Matolcsi. 2005

172 citations recorded by the source corpus.

In this note we modify a recent example of Tao and give an example of a set Omega subset of R-4 such that L-2(Omega) admits an orthonormal basis of exponentials {1/vertical bar Omega vertical bar 1/2 e(2 pi i )}(xi epsilon Lambda) for some set Lambda subset of R-4, but which does not tile R-4 by translations. This shows that one direction of Fuglede's conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/971177d887f74df4edb91116e248b6b96dd3b173

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Citations in the corpus

Analysis of orthogonality and of orbits in affine iterated function systems (2006, methodology): “tion” (a tile which is not a spectral set); speciﬁcally that sets which tile need not be spectral. This was then followed by Matolci’s improvement of Tao’s result down to four dimension, i.e., d = 4, [Mat06]. An example of a non-spectral tile has now been claimed for d = 3, [FMM06]. 2. Affine iterat”
Weak convergence and spectrality of infinite convolutions (2022, background): “Although the both directions of Fuglede’s conjecture were disproved by Tao [52] and the others [22, 23, 32, 33, 43] for d ≥ 3, the connection between spectrality and tiling has generated a lot of interest, and some affirmative results have been proved for special cases [29,34].”
Some reductions of the spectral set conjecture to integers (2013, background): “It was eventually shown that the conjecture is false in both directions on d 3 [3, 7, 8, 13].”
Tiles with no spectra (2004, background): “Tao [12] has recently proved that the direction ‘‘spectral ) tiling’’ does not hold (in dimension 5 and higher—Matolcsi [9] has reduced this dimension to 4).”
Tiles with no spectra in dimension 4 (2006, methodology): “Kolountzakis and Matolcsi [7] have applied this method with one-dimensional group homomorphisms ϕ : Z56 → Z6, in connection with a 5-dimensional decomposition of the matrix K .”
Complex Hadamard matrices and the spectral set conjecture (2004, methodology): “We can transfer this counterexample to Z and R in the same way as described in [16] and [13].”
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].”
Fuglede's conjecture on cyclic groups of order pnq (2016, result): “Subsequently, works by Kolountzakis and Matolcsi [9, 10, 14] and Farkas-Matolcsi-Mora [5] showed that the conjecture is false in dimensions 3 and above in both directions, leaving it open for R and R2.”