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 eλ(x) = exp(2πiλx), λ ∈ Λ, form a complete orthonormal system of L2(Ω). Such a set Λ is called a spectrum of Ω. In this note we prove that any spectrum Λ of a bounded measurable set Ω ⊆ R must be periodic.