Periodicity of the spectrum of a finite union of intervals

TL;DR

This paper provides a simplified proof that any spectrum of a finite union of intervals with Lebesgue measure 1 must be periodic, advancing understanding of spectral sets in harmonic analysis.

Contribution

The paper offers a more straightforward proof of the periodicity of spectra for finite unions of intervals, improving upon the original proof by Bose and Madan.

Findings

Spectra of finite unions of intervals are necessarily periodic.

The simplified proof clarifies the structure of spectral sets.

Supports the conjecture that spectral sets have regular patterns.

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.