Spectrum is rational in dimension one

Abstract: A bounded measurable set $\Omega\subset{\mathbb R}^d$ is called a spectral set if it admits some exponential orthonormal basis ${e^{2\pi i \langle\lambda,x\rangle}: \lambda\in\Lambda}$ for $L^2(\Omega)$. In this paper, we show that in dimension one $d=1$, any spectrum $\Lambda$ with $0\in\Lambda$ of a spectral set $\Omega$ with Lebesgue measure normalized to 1 must be rational. Combining previous results that spectrum must be periodic, the Fuglede's conjecture on ${\mathbb R}^1$ is now equivalent to the corresponding conjecture on all cyclic groups ${\mathbb Z}_{n}.$