We introduce a notion of -almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded -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.
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Svetlana Jitomirskaya, Shiwen Zhang, Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators. J. Eur. Math. Soc. 24 (2022), no. 5, pp. 1723–1767DOI 10.4171/JEMS/1139