TITLE: Lebesgue measure of a set of irrational numbers
QUESTION [10 upvotes]: Let $I_{\lambda},$ $\lambda>0$ be a subset of all irrational numbers $\rho=[a_{1},a_{2},...,a_{n},...]\in(0,1)$ such that $a_{n}\leq \text{const}\cdot n^{\lambda}.$
Here, $[a_{1},a_{2},...,a_{n},...]$ is the continued fraction with partial quotients $a_1,a_2,\dots$. 
My question is: under what values of $\lambda$ the set $I_{\lambda}$ has a positive Lebesgue measure?

REPLY [2 votes]: A precise solution to this problem is known. In Khintchine's book on continued fractions it says:
$\mathbf{Theorem~30}$ Suppose that $\varphi(n)$ is an arbitrary positive function with natural argument $n$. The inequality
$$
a_n = a_n(\alpha) \geq \varphi(n)
$$
is, for almost all $\alpha$, satisfied by an infinite number of values $n$ if the series $\sum_n 1/\varphi(n)$ diverges. On the other hand, this inequality is, for almost all $\alpha$, satisfied by only a finite number of values of $n$ if the series $\sum_n 1 /\varphi(n)$ converges.
(quoting from: A. Ya. Khintchine. Continued Fractions. University of Chicago Press, 1964)