On the large partial quotients in the continued fraction expansion (Q1434328)

This paper studies the metric behaviour of the large partial quotients in the continued fraction expansion; a large partial quotient stands for a partial quotient \(a_i\) for which \(a_i=\max(a_1,\cdots,a_i)\). More precisely, for a positive real number \(x= [a_0; a_1, \cdots, a_i, \cdots ]\) with unbounded partial quotients \(a_i\), let \((B_n(x))\) stand for the increasing sequence of large partial quotients in the continued fraction expansion of \(x\), and let \((T_n(x))\) denote the sequence of successive occurrences of those partial quotients: for all \(n\), one has \(B_n(x)=a_{T_n}(x)\). Special focus is given on the number \(N_n(x)\) of repetitions of the partial quotient \(B_n(x)\) in the range \((T_n(x), T_{n+1}(x))\), that is, before the apparition of the next large partial quotient. It is first proved that the length of the range \((T_n(x), T_{n+1}(x))\) may be very long, that is, for a.e. \(x\), \(\lim_{n \rightarrow \infty} \frac{T_{n+1}(x) - T_n (x)}{u ^n}=\infty\) for all \(u\) with \(1 < u <2\). Nevertheless, the main result of the paper is that there are no repetitions from some rank on, that is, for a.e. \(x\), there exists \(n_0(=n_0(x))\) such that \(N_n(x)=0\) for \(n \geq n_0\). Furthermore, it is proved that for a.e. \(x\), \(\lim_{ n \rightarrow \infty} \frac{B_n(x)}{u ^n}= \infty\) for all \( 1 < u <2\); the distribution of the repartition for the sequence \((\frac{a_n}{\max(a_1,\cdots,a_n)})\) is also studied. The proofs are elegantly written and based on the \(\psi\)-mixing of the process of partial quotients and on the sub-additive ergodic theorem for what concerns the repartition function for the sequence \((\frac{a_n}{\max(a_1,\cdots,a_n)})\). The results are numerically illustrated in the case of \(2 ^{1/3}\) and \(\pi\); the computations are based on [\textit{C. Faivre}, Acta Sci. Math. 67, 505--519 (2001; Zbl 1017.11040)] which provides the number of required decimals to obtain a prescribed number of partial quotients. Some compelling questions and conjectures are also stated.