On The Rationality Of The Spectrum

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 $\La...

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Bibliographic Details
Main Authors Bose, Debashish, Madan, Shobha
Format Journal Article
LanguageEnglish
Published 15.06.2016
Subjects
Mathematics - Classical Analysis and ODEs
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DOI10.48550/arxiv.1606.04814

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Summary: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.
DOI:10.48550/arxiv.1606.04814