Universal spectra, universal tiling sets and the spectral set conjecture

Abstract

A subset $\Omega$ of $\mathbf{R}^d$ with finite positive Lebesgue measure is called a spectral set if there exists a subset $\Lambda\subset\mathbf{R}$ such that ${\mathcal E}_\Lambda :=\{e^{i2\pi \langle\lambda, x\rangle}: \lambda\in\Lambda\}$ form an orthogonal basis of $L^2(\Omega)$. The set $\Lambda$ is called a spectrum of the set $\Omega$. The Spectral Set Conjecture states that $\Omega$ is a spectral set if and only if $\Omega$ tiles $\mathbf{R}^d$ by translation. In this paper we prove the Spectral Set Conjecture for a class of sets $\Omega \subset \mathbf{R}$. Specifically we show that a spectral set possessing a spectrum that is a strongly periodic set must tile $\mathbf{R}$ by translates of a strongly periodic set depending only on the spectrum, and vice versa.

On Fuglede's Conjecture and the Existence of Universal Spectra

“…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 in R 5 and higher dimensions.…”

Quasiperiodic spectra and orthogonality for iterated function system measures

“…The following result (Theorem 3.29) is closely related to one in [PW01], but we include the details here since our techniques are different. Specifically, we stress the twisted tensor product of Hadamard matrices (3.9); a computational feature motivated by fast Fourier transform algorithms for finite groups.…”

“…Hence in the literature, starting with [PW01,LW96], a number of authors have placed additional conditions on the sets in (2.1) and the spectra in (2.2) with view to more definite results. For example, in [Ped04a,Ped04b] Pedersen introduced an intriguing "dual spectral-set-conjecture".…”

Fuglede’s conjecture for a union of two intervals

“…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] for a discussion). See also [16], [1] for partial results on the related problem of characterizing all tilings of Z by a finite set, and [15], [18] for a classification of domains in R n which have L + Z n as a spectrum for some finite set L. Another recent result [19] is that sets which tile [0, ∞) by translations are spectral sets. The purpose of the present article is to address the following special case of Fuglede's conjecture in one dimension.…”