On The Rationality Of The Spectrum

TL;DR

This paper investigates the properties of spectral sets and their spectra, providing partial results that support the conjecture that spectra are rational and related to tiling properties, building on prior work on periodicity and rationality.

Contribution

The paper offers new partial results that support the conjecture that spectra of certain sets are rational and periodic, advancing understanding of the spectral set conjecture.

Findings

Spectra are shown to be periodic in certain cases

Evidence supporting the rationality of spectra is provided

Partial results relate spectra to tiling properties

Abstract

Let $\Omega \subset \mathbb{R}$ be a compact set with measure $1$. If there exists a subset $\Lambda \subset \mathbb{R}$ such that the set of exponential functions $E_{\Lambda}:=\{e_\lambda(x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in \Lambda\}$ is an orthonormal basis for $L^2(\Omega)$, then $\Lambda$ is called a spectrum for the set $\Omega$. A set $\Omega$ is said to tile $\mathbb{R}$ if there exists a set $\mathcal T$ such that $\Omega + \mathcal T = \mathbb{R}$. A conjecture of Fuglede suggests that Spectra and Tiling sets are related. Lagarias and Wang \cite {LW1} proved that Tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in \cite {BM1}, \cite {IK}. In this paper, we give some partial results to support the rationality of the spectrum.