Periodicity of the spectrum in dimension one

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

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