On Fuglede's Conjecture and the Existence of Universal Spectra

By Bálint Farkas, Máté Matolcsi, Péter Móra. 2006

170 citations recorded by the source corpus.

Recent methods developed by Tao [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde’s Spectral Set Conjecture in both directions. Namely, in ${\Bbb R}^5$ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a nonspectral tile. In search of lower dimensional nonspectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang [14]. In particular, we prove here that the USC and the "tile → spectral" direction of Fuglede’s conjecture are equivalent in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szabó [13] for the existence of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples of tiles which have no spectra. We overcome these difficulties by invoking some ideas of Révész and Farkas [2], and obtain nonspectral tiles in ${\Bbb R}^3$. Fuglede’s conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The one-dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz [1].

Subfields: Applied Mathematics, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c26404f9fdb5518e6dd7ef64d6b0df7d43eff05a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Materials Chemistry

Mihail N. Kolountzakis

In Mathematical Physics

Citations in the corpus

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.”
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].”
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].”
Fuglede's conjecture on cyclic groups of order pnq (2016, background): “cture for dimensions 5 and above [15], constructing a spectral set in 5 dimensions that does not tile the space1. 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 ope”
Analysis of orthogonality and of orbits in affine iterated function systems (2006, background): “An example of a non-spectral tile has now been claimed for d = 3, [ FMM06 ].”