# Computational results related to Khinchin-Levy constants

Khinchin's Theorem in Diophantine approximations state that for a 'typical' real number $\alpha$ (all but a set of Lebesgue measure 0), the continued fraction $\alpha = [a_0, a_1, \cdots]$ where $a_0 > 0$, satisfy $\displaystyle \lim_{n \rightarrow \infty} (a_0 \cdots a_n)^{1/n} = K_0$ for some constant $K_0$, known as Khinchin's constant. Similarly, Levy in 1935 proved that for almost all real $\alpha = [a_0, a_1, \cdots ]$ and $p_n/q_n = [a_0, \cdots, a_n]$ we have $\displaystyle \lim_{n \rightarrow \infty} q_n^{1/n} = e^{\pi^2/12 \log(2)}$, where the latter is known as Levy's constant. It is not known to verify whether a given number is a 'typical' real number (in the sense of the above two theorems) efficiently. I am wondering are there any results stating how fast or slow the 'typical' real number ought to converge to the two limit values. For example, what is the expected size of $n$ such that $|q_n^{1/n} - e^{\pi^2/12 \log(2)}| < 0.01$ for a typical $\alpha$?

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this is standard once you understand the ergodic theory behind the Gauss map.You are precisely asking about the rate of convergence in the ergodic theorem for a specific test function. The Central Limit Theorem applies in a nice way. You can get bounds from there. –  Anthony Quas Feb 15 '11 at 21:45