Given a periodic Schrödinger operator with analytic potential and rational frequency α, let S_- denote the intersection of its spectra taken over the phase x in a torus. We show that up to sets of Lebesgue measure zero, S_- associated with α could be obtained asymptotically from S_- associated with rationals approximating α that satisfy certain approximating properties. We will talk more about the proof details. This work is joint with Svetlana Jitomirskaya and Xianzhe Li. This talk is part of the WSMP program.