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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

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If you have access to Math Reviews, you'll find that MR0637846 (83d:10063) Misyavichyus, G. A., Estimate of the remainder term in the limit theorem for the denominators of continued fractions. (Russian. English, Lithuanian summary) Litovsk. Mat. Sb. 21 (1981), no. 3, 63–74, is something like what you want.

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