Let Ω ⊂ R be a compact set with measure 1. If there exists a subset Λ ⊂ R such that the set of exponential functions EΛ: = { eλ(x) = e2 π i λ x| Ω: λ∈ Λ } is an orthonormal basis for L2(Ω) , then Λ is called a spectrum for the set Ω. A set Ω is said to tile R if there exists a set T such that Ω + T= R, the set T is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013). In this paper, we give some partial results to support the rationality of the spectrum. © 2017, Springer Science+Business Media, LLC.