Universal spectra, universal tiling sets and the spectral set conjecture - Pedersen, Steen; Wang, Yang (Mathematica Scandinavica, 2001)
A subset $Ω$ of $R^d$ with finite positive Lebesgue measure is called a spectral set if there exists a subset $Λ\subsetR$ such that ${\mathcal E}_Λ :=\{e^{i2π \langleλ, x\rangle}: λ\inΛ\}$ form an orthogonal basis of $L^2(Ω)$. The set $Λ$ is called a spectrum of the set $Ω$. The Spectral Set Conjecture states that $Ω$ is a spectral set if and only if $Ω$ tiles $R^d$ by translation. In this paper we prove the Spectral Set Conjecture for a class of sets $Ω \subset R$. Specifically we show that a spectral set possessing a spectrum that is a strongly periodic set must tile $R$ by translates of a strongly periodic set depending only on the spectrum, and vice versa.