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.