On the Rationality of the Spectrum

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 $\Omega + \mathcal T = {\mathbb R}$, the set $\mathcal T$ is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013). In this paper, we give some partial results to support the rationality of the spectrum.