Spectrum is rational in dimension one

TL;DR

This paper proves that in one-dimensional space, any spectrum of a spectral set with measure 1 must be rational, linking the spectral set conjecture to properties of cyclic groups.

Contribution

It establishes the rationality of spectra in one dimension, connecting spectral set conjectures to cyclic group properties.

Findings

Spectra in 1D spectral sets are rational numbers.

Spectra must be periodic in 1D.

Fuglede's conjecture in 1D reduces to cyclic group conjectures.

Abstract

A bounded measurable set $\Omega\subset{\mathbb R}^d$ is called a spectral set if it admits some exponential orthonormal basis $\{e^{2\pi i \langle\lambda,x\rangle}: \lambda\in\Lambda\}$ for $L^2(\Omega)$. In this paper, we show that in dimension one $d=1$, any spectrum $\Lambda$ with $0\in\Lambda$ of a spectral set $\Omega$ with Lebesgue measure normalized to 1 must be rational. Combining previous results that spectrum must be periodic, the Fuglede's conjecture on ${\mathbb R}^1$ is now equivalent to the corresponding conjecture on all cyclic groups ${\mathbb Z}_{n}.$