  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 $\beta$-H\"older continuity of the
integrated density of states for supercritical quasi-periodic Schr\"odinger
operators restricted to the $\ell$-th stratum, for any $\beta<(2(\ell-1))^{-1}$
and $\ell\ge2$. 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.
