  Let $\Omega \subset \mathbb{R}$ be a compact set with measure $1$. If there
exists a subset $\Lambda \subset \mathbb{R}$ such that the set of exponential
functions $E_{\Lambda}:=\{e_\lambda(x) = e^{2\pi i \lambda x}|_\Omega :\lambda
\in \Lambda\}$ is an orthonormal basis for $L^2(\Omega)$, then $\Lambda$ is
called a spectrum for the set $\Omega$. A set $\Omega$ is said to tile
$\mathbb{R}$ if there exists a set $\mathcal T$ such that $\Omega + \mathcal T
= \mathbb{R}$. 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.
