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 E Λ := {e λ (x) = e 2πiλx | Ω : λ ∈ Λ} is an orthonormal basis for L 2 (Ω), then Λ is called a spectrum for the set Ω. A set Ω is said to tile R if there exists a set T such that Ω + T = R. A conjecture of Fuglede suggests that Spectra and Tiling sets are related. Lagarias and Wang [14] proved that Tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in[3], [8]. In this paper, we give some partial results to support the rationality of the spectrum.

Spectrum is rational in dimension one

“…It has been conjectured for quite some time that any spectrum for a spectral set Ω in R 1 with |Ω| = 1 must be rational as well. A partial result of this conjecture was proved by Bose and Madan [1] for very special Ω's.…”