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 ΛR such that E_Λ :={e^i2π λ, x: λΛ} 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 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.