Spectrum is rational in dimension one

Abstract

A bounded measurable set Ω ⊂ R d is called a spectral set if it admits some exponential orthonormal basis {e 2πi λ,x : λ ∈ Λ} for L 2 (Ω). In this paper, we show that in dimension one d = 1, any spectrum Λ with 0 ∈ Λ of a spectral set Ω with Lebesgue measure normalized to 1 must be rational. Combining previous results that spectrum must be periodic, the Fuglede's conjecture on R 1 is now equivalent to the corresponding conjecture on all cyclic groups Zn.