Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators

Abstract

We introduce a notion of $\beta$-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded $\beta$-almost periodic potentials. Applications include the first sharp arithmetic spectral criterion for the entire family of supercritical analytic quasiperiodic Schrödinger operators and arithmetic spectral/quantum dynamical criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator. In particular, we disprove a 1994 conjecture of Wilkinson–Austin.