Complex Hadamard matrices and the spectral set conjecture

By Mihail N. Kolountzakis, Máté Matolcsi. 2004

219 citations recorded by the source corpus.

By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction ?spectral ) tile? of the Spectral Set Conjecture, for all sets A of size |A| 5, in any finite Abelian group. This result is then extended to the infinite grid Zd for any dimension d, and finally to Rd. It was pointed out recently in [16] that the corresponding statement fails for |A| = 6 in the group Z5 3, and this observation quickly led to the failure of the Spectral Set Conjecture in R5 [16], and subsequently in R4 [13]. In the second part of this note we reduce this dimension further, showing that the direction ?spectral ) tile? of the Spectral Set Conjecture is false already in dimension 3.

Subfields: Algebra and Number Theory, Applied Mathematics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/5a262f27ce45f607c0b1616c9db6a3a1594092dc

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Citations in the corpus

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].”
Multiple lattice tiles and Riesz bases of exponentials (2013, background): “The so called Fuglede or Spectral Set Conjecture [4] (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 [20, 12, 2, 3, 11], in both directions.”
Construction, classification and parametrization of complex Hadamard matrices (2011, background): “Tao’s arguments also implied that Fuglede’s conjecture fails in R in any dimension higher than 5 and subsequent counterexamples were found refuting both directions for d ≥ 3, see [49] and [84].”
On Fuglede's Conjecture and the Existence of Universal Spectra (2006, methodology): “Kolountzakis and Matolcsi [8] reduced this dimension to 3, and also constructed a counterexample to the “tile → spectral” direction in R(5), see [7].”
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].”
Spectral property of Cantor measures with consecutive digits (2012, background): “ture baﬄed experts for 30 years until 2004, Tao [T] gave the ﬁrst counterexamples on Rd, d ≥ 5. The examples were modiﬁed later so that the conjecture are false in both directions on Rd, d ≥ 3 [KM1], [KM2]. It remains open in dimension 1 and 2. Despite the counterexamples, the exact relationship bet”
Spectrality of Self-Similar Tiles (2013, result): “cience Foundation of China 11271148. 1 The conjecture was ﬁnally disproved by Tao [T] that a spectra l set is not necessary a tile for Rs,s ≥ 5, and subsequently improved by Kolountzakis and Matolcsi [KM1,2] that the conjecture is false in both directions for s ≥ 3. Nevertheless, the conjecture is s”
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].”
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.”
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”