Periodicity of the spectrum in dimension one

TL;DR

This paper proves that any spectrum associated with a bounded measurable set of measure 1 in the real line must be periodic, revealing a fundamental property of spectral sets in one dimension.

Contribution

It establishes the periodicity of spectra for all bounded measurable spectral sets in one dimension, a significant advancement in spectral set theory.

Findings

Spectra of bounded measurable sets in one dimension are necessarily periodic.

The result applies to all sets of Lebesgue measure 1.

This confirms a key structural property of spectral sets in the real line.

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