Homogeneity of the spectrum for quasi-periodic Schrodinger operators

We consider the one-dimensional discrete Schrodinger operator [H(x,omega)phi](n) -phi(n - 1) -phi(n + 1) + V(x + n omega)phi(n), n is an element of Z, x, omega is an element of [0, 1], with real-analytic potential V(x). Assume L(E, omega) > 0 for all E. Let S-omega be the spectrum of H(x, omega). For all omega obeying the Diophantine condition omega is an element of T-c,T-a, we show the following: if S-omega boolean AND (E', E '') not equal empty set, then S-omega boolean AND (E', E '') is homogeneous in the sense of Carleson [Car83]. Furthermore, we prove that if G(i), i = 1, 2, are two gaps with 1 > vertical bar G(1)vertical bar >= vertical bar G(2)vertical bar, then vertical bar G(2)vertical bar less than or similar to exp (- (log dist(G(1), G(2)))(A)), A >> 1. Moreover, the same estimates hold for the gaps in the spectrum on a finite interval, that is, for S-N,S-omega := boolean OR(x is an element of T) spec H-[-N,H-N] (x, omega), N >= 1, where H-[-N,H- N](x, omega) is the Schrodinger operator restricted to the interval [-N, N] with Dirichlet boundary conditions. In particular, all these results hold for the almost Mathieu operator with vertical bar lambda vertical bar not equal 1. For the supercritical almost Mathieu operator, we combine the methods of [GS08] with Jitomirskaya's approach from [Jit99] to establish most of the results from [GS08] with omega obeying a strong Diophantine condition.