On The Rationality Of The Spectrum - Bose, Debashish; Madan, Shobha (2016)
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 $E_{Λ}:=\{e_λ(x) = e^{2π i λ x}|_Ω :λ \in Λ\}$ 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.