Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

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