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On The Rationality Of The Spectrum

Let $Ω\subset \mathbb{R}$ be a compact set with measure $1$. If there exists a subset $Λ\subset \mathbb{R}$ such that the set of exponential functions $E_Λ:=\{e_λ(x) = e^{2πi λx}|_Ω:λ\in Λ\}$ is an orthonormal basis for $L^2(Ω)$, then $Λ$ is called a spectrum for the set $Ω$. A set $Ω$ is said to tile $\mathbb{R}$ if there exists a set $\mathcal T$ such that $Ω+ \mathcal T = \mathbb{R}$. A conjecture of Fuglede suggests that Spectra and Tiling sets are related. Lagarias and Wang \cite {LW1} proved that Tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in \cite {BM1}, \cite {IK}. In this paper, we give some partial results to support the rationality of the spectrum.