The absolutely continuous spectrum of finitely differentiable quasi-periodic Schr\"{o}dinger operators

We prove that the quasi-periodic Schr\"{o}dinger operator with a finitely differentiable potential has purely absolutely continuous spectrum for all phases if the frequency is Diophantine and the potential is sufficiently small in the corresponding $C^k$ topology. This is based on a refined quantitative $C^{k,k_0}$ almost reducibility theorem which only requires a quite low initial regularity ``$k>14\tau+2$'' and much of the regularity ``$k_0\leq k-2\tau-2$'' is conserved in the end, where $\tau$ is the Diophantine constant of the frequency.