On the density of states of the one dimensional quasi-periodic Schrodinger operators

We prove two results on the density of states of the discrete one dimensional quasi-periodic Schrodinger equation with an analytic potential and Diophantine frequencies in the perturbed regime. On the one hand, we prove that this function has the behavior of a Holder-1/2 function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem.