Avila's acceleration via zeros of determinants, and applications to Schrödinger cocycles

In this paper we give a characterization of Avila’s quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove $\beta$-Hölder continuity of the integrated density of states for supercritical quasi-periodic Schrödinger operators restricted to the $\mathrm{\ell }$-th stratum, for any $\beta <(2(\mathrm{\ell }-1){)}^{-1}$ and $\mathrm{\ell }\ge 2$. We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty.