Fuglede’s conjecture for a union of two intervals

By Izabella Laba. 2000

149 citations recorded by the source corpus.

We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/88932560e91159c2db08103d268ac2a174378567

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 Mathematical Physics

Citations in the corpus

On Fuglede's Conjecture and the Existence of Universal Spectra (2006, result): “of the conjecture, when the spectrum or the tiling complement is assumed to be a lattice. Later on, for many years several positive results seemed to indicate the validity of the conjecture (see e.g. [5, 4, 6, 9, 10, 11, 13, 15]. Recently, however, Tao [17] disproved the “spectral → tile” direction ”
Some properties of spectral measures (2006, unknown): “Lagarias and Wang [LaWa97b], Pedersen and Wang [PeWa01], Laba [ Lab01 ], and Iosevich, Katz and Tao [IKT03].”
Fuglede's conjecture on cyclic groups of order pnq (2016, background): “ng complement of Ω is a lattice. Later on, the conjecture was proved for various families of subsets by numerous authors, for example, for 2-dimensional convex bodies [7], unions of two intervals in R[11], etc. Against the current of research on this subject, Tao disproved this conjecture for dimens”
The Spectral Set Conjecture and Multiplicative Properties of Roots of Polynomials (2002, background): “Note that the set [0, n] ∪ [2n, 3n] tiles R and is a spectral set (see for example [17] for an explicit calculation of the spectra), but is not a fundamental domain for any group; this shows that the inequality M < 3N/2 cannot be weakened.”
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].”
Fuglede’s conjecture fails in dimension 4 (2005, background): “…will restrict our attention to sets of the form of a finite union of unit cubes placed at points with integer coordinates, i.e., (1) Ω = ⋃ t∈Γ {t + [0, 1)d}, where Γ ⊂ Zd is finite (this means, essentially, that we will be considering the tiling and spectral properties of sets in Zd).”
Multi-tiling and Riesz bases (2012, background): “For example, it is easy to construct a union of two intervals on R which does not admit such a basis (see also [13]).”
Construction, classification and parametrization of complex Hadamard matrices (2011, methodology): “Several positive partial results were obtained regarding some special cases of Fuglede’s conjecture (see [69], [94]), before Tao exhibited a non-tiling spectral set in Z(6)3, which then he slightly modified to obtain a counterexample in Z(5)3 as well.”
Spectrality of Self-Similar Tiles (2013, background): “n the last four decades. There were many partial results, and involved a lot of techniques from harmonic analysis, tiling theory, convex geometry and geometry of numbers ([P], [LW4], [IKP], [IKT1,2], [Ł1,2]). 1991 Mathematics Subject Classiﬁcation. Primary 42C15; Secondary 28A80. Key words and phras”
When does a Bernoulli convolution admit a spectrum (2012, background): “On the other hand, strong connection between spectral sets and tiling has been observed especially in low dimensions; see [1,12,13,17,21,23] and the survey [18].”