An ergodic sum related to the approximation by continued fractions (Q2581080)

Let \(x \in \mathbb R \setminus \mathbb Q\) and let \(p_n/q_n\) denote the \(n\)th convergent of \(x\) in the usual simple continued fraction expansion of \(x\). Define \[ \theta_n(x) = q_n^2 \left| x - {{p_n}\over{q_n}}\right|. \] From a classical theorem of Dirichlet, it follows that \(\theta_n(x) \in (0,1)\). In the present paper, it is shown that for almost all \(x \in \mathbb R\), \[ \lim_{n \rightarrow \infty} {{1}\over{n}} \sum_{k=1}^n \log \theta_k(x) = -1-{{1}\over{2}}\log 2. \] The main result is derived by considering a natural extension of the Gauss map, which turns out to be ergodic due to a result of \textit{H. Nakada} [Tokyo J. Math. 4, 399--426 (1981; Zbl 0479.10029)]. The result is then proved by applying the Birkhoff ergodic theorem to a suitable function and showing that the conclusion remains valid under projection back to the original system.