It is well known, by the Birkhoff's pointwise ergodic theorem, that the
time average (ergodic average) converges to the space average
(expectation) for almost all points in an ergodic dynamical system. What
can be said about the rate of the convergence?
In this talk, we shall derive an error term of the difference between
the ergodic average and the expectation for some ergodic dynamical
systems under certain condition. The required condition involves an upper
bound for the $L^2$-convergence of the ergodic average. In particular,
this requirement can be attained by the strong mixing condition
($\alpha$-mixing property) from the viewpoint of stochastic processes.
For example, it is known that the regular continued fractions satisfy
the $\psi$-mixing property, which implies the strong mixing condition,
so we can apply our quantitative ergodic theorem to refine classical
results of the metric theory of continued fractions. That is, we shall
see that the geometric mean of the partial quotients of the continued
fraction expansion converges to the Khinchin's constant with an error
term of order $o(N^{-1/2}(\log N)^{3/2}(\log(\log N))^{1/2 +\epsilon})$.
It is worth noting that our method gives an improvement on the error term
derived from the classical method of I.S. Gal and J.F. Koksma (1950).