I'm wondering if anyone knows of a reference other than Khinchin's Metrische Kettenbruchproblem http://www.numdam.org/item/CM_1935__1__361_0/ for the following result of Khinchin mentioned in a paper of Diamond and Vaaler (p.74) https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-122/issue-1/Estimates-for-partial-sums-of-continued-fraction-partial-quotients/pjm/1102702122.full
\begin{align}(b_1 + \cdots + b_N)/(N\log{N}) \to 1/\log{2} \end{align} in measure as $N \to \infty$, where \begin{align} b_n = \begin{cases} a_n, \text{ if } a_n < n(\log{n})^{4/3} \\\\ 0, \text{ otherwise. } \end{cases} \end{align}
Here, $a_n$ is the $n-th$ digit or partial quotient of the continued fraction expansion of an irrational number in $(0,1)$.
Thanks in advance!