On the Rationality of the Spectrum

Abstract

Let Ω⊂ℝ be a compact set with measure 1. If there exists a subset Λ⊂ℝ such that the set of exponential functions 𝐸Λ:={𝑒𝜆(𝑥)=𝑒2𝜋𝑖𝜆𝑥|Ω:𝜆∈Λ} is an orthonormal basis for 𝐿2(Ω), then Λ is called a spectrum for the set Ω. A set Ω is said to tile ℝ if there exists a set  such that Ω+=ℝ, the set  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.