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 $\{e^{2\pi i⟨\lambda ,x⟩}:\lambda \in \Lambda \}$ 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 ${\mathbb{Z}}_n.$

A gap was found in Section 4 of the paper, which appears to be uneasy to be resolved. We would like to thank Nir Lev for pointing it out. As a result, we would like to withdraw it now. Results before Section 4 are correct, we welcome someone to fix the gap later