Central Spectral Gaps of the Almost Mathieu Operator

We consider the spectrum of the almost Mathieu operator H-alpha with frequency alpha and in the case of the critical coupling. Let an irrational alpha be such that vertical bar alpha - pn/qn vertical bar < cq(n)(-x), where p(n)/q(n), n = 1, 2,... are the convergents to alpha, and c, x are positive absolute constants, x < 56. Assuming certain conditions on the parity of the coefficients of the continued fraction of alpha, we show that the central gaps of Hp(n)/q(n), n = 1, 2,..., are inherited as spectral gaps of H-alpha of length at least c'q(n)(-x/2) , c' > 0.