Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

We establish a quantitative version of strong almost reducibility result for $\mathrm{s}\mathrm{l}(2,\mathbb{R})$ quasi-periodic cocycle close to a constant in Gevrey class. We prove that, for the quasi-periodic Schr"odinger operators with small Gevrey potentials, the length of spectral gaps decays sub-exponentially with respect to its labelling, the long range duality operator has pure point spectrum with sub-exponentially decaying eigenfunctions for almost all phases and the spectrum is an interval for discrete Schr"odinger operator acting on ${\mathbb{Z}}^d$ with small separable potentials. All these results are based on a refined KAM scheme, and thus are perturbative.