Universal spectra, universal tiling sets and the spectral set conjecture

A subset Ω of R^d with finite positive Lebesgue measure is called a spectral set if there exists a subset Λ ⊂ R such that ℰ_Λ:= {ei2π(λ, x) : λ ∈ Λ} form an orthogonal basis of L²(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.