  For non-critical almost Mathieu operators with Diophantine frequency, we
establish exponential asymptotics on the size of spectral gaps, and show that
the spectrum is homogeneous. We also prove the homogeneity of the spectrum for
Sch\"odinger operators with (measure-theoretically) typical quasi-periodic
analytic potentials and fixed strong Diophantine frequency. As applications, we
show the discrete version of Deift's conjecture \cite{Deift, Deift17} for
subcritical analytic quasi-periodic initial data and solve a series of open
problems of Damanik-Goldstein et al \cite{BDGL, DGL1, dgsv, Go} and Kotani
\cite{Kot97}.
