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 $|\alpha-p_n/q_n|<c q_n^{-\varkappa}$, where $p_n/q_n$, $n=1,2,\dots$ are the convergents to $\alpha$, and $c$, $\varkappa$ are positive absolute constants, $\varkappa<56$. Assuming certain conditions on the parity of the coefficients of the continued fraction of $\alpha$, we show that the central gaps of $H_{p_n/q_n}$, $n=1,2,\dots$, are inherited as spectral gaps of $H_\alpha$ of length at least $c'q_n^{-\varkappa/2}$, $c'>0$.