Periodicity of the spectrum in dimension one

Abstract

A bounded measurable set $\Omega$, of Lebesgue measure 1, in the real line is called spectral if there is a set $\Lambda$ of real numbers ("frequencies") such that the exponential functions $e_{\lambda }(x)=\exp (2\pi i\lambda x)$, $\lambda \in \Lambda$, form a complete orthonormal system of $L^2(\Omega )$. Such a set $\Lambda$ is called a {\em spectrum} of $\Omega$. In this note we prove that any spectrum $\Lambda$ of a bounded measurable set $Ω\subseteq\RR$ must be periodic.

Correction of an error pointed out by Dorin Dutkay; Lemma 1 now has a new proof