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Spectrum is rational in dimension one

A bounded measurable set $Ω\subset{\mathbb R}^d$ is called a spectral set if it admits some exponential orthonormal basis $\{e^{2πi \langleλ,x\rangle}: λ\inΛ\}$ for $L^2(Ω)$. In this paper, we show that in dimension one $d=1$, any spectrum $Λ$ with $0\inΛ$ 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 ${\mathbb R}^1$ is now equivalent to the corresponding conjecture on all cyclic groups ${\mathbb Z}_{n}.$