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 $e_{\lambda }\left(x\right)=exp\left(2\pi i\lambda x\right)$, $\lambda \in \Lambda$, form a complete orthonormal system of $L^2\left(\Omega \right)$. Such a set $\Lambda$ is called a spectrum of $\Omega$. In this note we prove that any spectrum $\Lambda$ of a bounded measurable set $\Omega \subseteq ℝ$ must be periodic.