On The Rationality Of The Spectrum

Abstract

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.