A subset Ω of R~d with finite positive Lebesgue measure is called a spectral set if there exists a subset Λ is contained in R such that ε_Λ := {e~(i2π<λ,x>):λ ∈ Λ} 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 is contained in 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.
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