  In this paper, we demonstrate a polynomial convergence rate for
homogenization of Hamilton-Jacobi equations with quasi-periodic potentials. We
establish a connection between the convergence rate of homogenization and the
regularity of the effective Hamiltonian, by using a new quantitative ergodic
estimate for bounded quasi-periodic functions with Diophantine frequencies. As
an application, we also study the convergent rate for Birkhoff average of
unbounded quasi-periodic functions.
