Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

We establish a quantitative version of strong almost reducibility result for SL(2, R) quasi-periodic cocycle close to a constant in the Gevrey class. We prove that, if the frequency is Diophantine, the long range operator has pure point spectrum with sub-exponentially decaying eigenfunctions for almost all phases; for the one dimensional quasi-periodic Schrodinger operators with small Gevrey potentials, the length of spectral gaps decays sub-exponentially with respect to its labelling; and the spectrum is an interval for discrete Schrodinger operators acting on Z(d) with small separable potentials.