Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys

This paper establishes several sharp spectral results for analytic quasiperiodic Schrodinger operators. Key contributions include: (1) exact exponential decay rates for spectral gaps of the almost Mathieu operator, addressing a question raised by Goldstein; (2) sharp arithmetic Anderson localization for a class of quasiperiodic operators on higher-dimensional lattices, which in particular resolves and generalizes Jitomirskaya's phase transition conjecture; and (3) stratified growth patterns for extended eigenfunctions revealing universal partial hierarchical structures for subcritical quasiperiodic Schrodinger operators. The proofs are based on novel frameworks-structured quantitative almost reducibility and sharp quantitative duality-to overcome the longstanding challenge of taming infinitely many rotation-number resonances, which enables us to obtain optimal arithmetic reducibility results for analytic SL(2,R)-cocycles, thereby solving another Jitomirskaya's conjecture. These methods enable a first comprehensive treatment of resonance-driven dynamical asymptotics.