Periodicity of the spectrum of a finite union of intervals

Abstract

A set $\Omega$, of Lebesgue measure 1, in the real line is called spectral if there is a set $\Lambda$ of real numbers such that the exponential functions $e_\lambda(x) = \exp(2\pi i \lambda x)$ form a complete orthonormal system on $L^2(\Omega)$. Such a set $\Lambda$ is called a spectrum of $\Omega$. In this note we present a simplified proof of the fact that any spectrum $\Lambda$ of a set $\Omega$ which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.