The statistics of the continued fraction digit sum (Q1568912)

Let \(\alpha\in (0,1)\) be an irrational number, and let \(a_k(\alpha)\) be the \(k\)th partial quotient in the regular continued fraction expansion of \(\alpha\). The main result of this paper is a limit theorem which tells us that the Lebesgue measure \(m(.)\) of the sets \[ A_n(z)= \Biggl\{ \alpha\in (0,1): \frac{a_1(\alpha)+\cdots+ a_n(\alpha)}{n}- \log n+ \gamma+ \log\log 2\leq z\Biggr\} \] converges (as \(n\to\infty\)) to a stable distribution function \(G(z)\) with characteristic exponent 1 and skewness parameter 1. Here \(\gamma\) denotes the Euler-Mascheroni constant, which is not mentioned in the paper. It should be mentioned that this result has been already proved in an earlier work of the reviewer published in 1987 [see \textit{L. Heinrich}, Math. Nachr. 131, 149--165 (1987; Zbl 0631.60020)], where, slightly different from the author's result, instead of \(m(.)\) the Gauss measure was used, which even provides an optimal rate of convergence.