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We prove sharp spectral transition in the arithmetics of phase betweenlocalization and singular continuous spectrum for Diophantine almost Mathieuoperators. We also determine exact exponential asymptotics of eigenfunctionsand of corresponding transfer matrices throughout the localization region. Thisuncovers a universal structure in their behavior governed by the exponentialphase resonances. The structure features a new type of hierarchy, whereself-similarity holds upon alternating reflections.

Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase