Polynomial decay of the gap length for C^k quasi-periodic Schrodinger operators and the spectral application

For C^k quasi-periodic Schrodinger operators in the local perturbative regime, we prove that the length of the corresponding spectral gap has a polynomial decay upper bound w.r.t. its label. This is based on a refined quantitative reducibility theorem for C^k quasi-periodic SL(2,R) cocycles. As an application, we are able to show the homogeneity of the spectrum.