Consider a quasi-periodic Schr\"odinger operator
$H_{\alpha,\theta}$ with analytic potential and Diophantine frequency
$\alpha$. Given any rational approximating $\alpha$, let $S_+$ and $S_-$
denote the union, respectively, the intersection of the spectra taken over
$\theta$. We show that up to sets of zero Lebesgue measure, the absolutely
continuous spectrum can be obtained asymptotically from $S_-$ of the
periodic operators associated with the continued fraction expansion of
$\alpha$. Similarly, from the asymptotics of $S_+$, one recovers the
spectrum of $H_{\alpha,\theta}$ (modulo a set of zero Lebesgue measure).