Tiles with no spectra

By Mihail N. Kolountzakis, Máté Matolcsi. 2004

270 citations recorded by the source corpus.

Abstract We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their L 2 space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang [Lagarias J. C. and Wang Y.: Spectral sets and factorizations of finite Abelian groups.J. Func. Anal. 145 (1997), 73–98]. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. We extend this last example to the groups ℤ d and ℝ d (for d ≥5 ) thus disproving one direction of the Spectral Set Conjecture of Fuglede [Fuglede B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16 (1974), 101–121]. The other direction was recently disproved by Tao [Tao T.: Fuglede's conjecture is false in 5 and higher dimensions. Math. Res. Letters 11 (2004), 251–258].

Subfields: Applied Mathematics, Mathematical Physics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/749b723aa4c4b587e73675a18493a1880966c420

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Citations in the corpus

Complex Hadamard matrices and the spectral set conjecture (2004, background): “HARP. 1 2 MIHAIL N. KOLOUNTZAKIS & MAT´ ´E MATOLCSI the ideas of [16] and [13] in order to reduce the dimension to 3. We also remark that the “tile ⇒ spectral” direction was recently disproved in [9] in dimension 5 and higher. The existing counterexamples do not exclude the possibility that the ”
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].”
Spectrality of Self-Similar Tiles (2013, background): “The conjecture was finally disproved by Tao [42], who showed that a spectral set is not necessary a tile forRs, s ≥ 5, and subsequently modified and proved by Kolountzakis and Matolcsi [25,26], who showed that the conjecture is false in both directions for s ≥ 3.”
When does a Bernoulli convolution admit a spectrum (2012, background): “It was proved to be false by Tao and others in dimension three or higher [19,20,22,31], but it is still open in dimension one and two.”
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].”
On spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem (2019, background): “l measures was originated from Fuglede [22], whose famous conjecture asserted that χΩdxis a spectral measure if and only if Ω is a translational tile. Although the conjecture was disproved eventually [31, 32, 42], the problem has led to the development of many new research problems and determining w”
Multiple lattice tiles and Riesz bases of exponentials (2013, background): “The so called Fuglede or Spectral Set Conjecture [3] (claiming that for Ω to have such a basis it is necessary and sufficient that it can tile space by translations) was eventually proved to be false in dimension at least 3 [15, 8, 1, 2], in both directions.”