Fuglede’s conjecture for a union of two intervals

Abstract

Abstract. We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations. The results A Borel set Ω ⊂ Rn of positive measure is said to tile R n by translations if there is a discrete set T ⊂ R n such that, up to sets of measure 0, the sets Ω + t, t ∈ T, are disjoint and t∈T (Ω + t) = R n . We may rescale Ω so that |Ω| = 1. We say that Λ = {λ k : k ∈ Z} ⊂ R n is a spectrum for Ω ifA spectral set is a domain Ω ⊂ R n such that (1.1) holds for some Λ. Fuglede [2] conjectured that a domain Ω ⊂ R n is a spectral set if and only if it tiles R n by translations, and proved this conjecture under the assumption that either Λ or T is a lattice. The conjecture is related to the question of the existence of commuting self-adjoint extensions of the operators −i Recently there has been significant progress on the special case of the conjecture when Ω is assumed to be convex [10], [3], [4], and in particular the 2-dimensional convex case appears to be nearly resolved [5]. The non-convex case is considerably more complicated, and is not understood even in dimension 1. The strongest results yet in that direction seem to be those of Lagarias and Wang [14], [15], who proved that all tilings of R by a bounded region must be periodic, and that the corresponding translation sets are rational up to affine transformations. This in turn leads to a structure theorem for bounded tiles. It was also observed in [15] that the "tiling implies spectrum" part of Fuglede's conjecture for compact sets in R would follow from a conjecture of Tijdeman [20] concerning factorization of finite cyclic groups; however, Tijdeman's conjecture is now known to fail without additional assumptions (see [13]

Some properties of spectral measures

“…Our next result concerns the density of A − A. Part (1) of it is a refinement of an argument in [Lab01].…”

“…Many positive results have been established as well, see e.g. Lagarias and Wang [LaWa97b], Pedersen and Wang [PeWa01], Laba [Lab01], and Iosevich, Katz and Tao [IKT03]. Interestingly, there is even evidence showing a strong connection between tiling and spectral measures, see [LabWa02] and Strichartz [Str00].…”

Riesz bases of exponentials and the Bohr topology

“…To be precise, a set Ω tiles R d by translations along a lattice if and only if the dual lattice is a spectrum for Ω. On the other hand, the conjecture is false in dimensions greater than two and some simple domains are not spectral; see [6,14,15,17] for examples.…”

Spectrality of a class of infinite convolutions

“…In another direction, Jorgensen and Pedersen [16] made a head start to study the spectral property of the self-similar measures. Nowadays, there is a large literature on this topic [1][2][3][4][5][6][7]9,[12][13][14][15][16][19][20][21][23][24][25][26][28][29][30]. Among those, one of the best known results is that if ρ = 1/q for some integer q > 1, then μ ρ,{0,1} is a spectral measure if and only if q is an even integer [16].…”