On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions (Q432597)

The authors consider continued fractions NEWLINE\[NEWLINE[a_1,a_2,\ldots]=\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\cdots}}}\in[0,1]NEWLINE\]NEWLINE and study properties of sum level sets NEWLINE\[NEWLINE\mathcal C_n=\{[a_1,a_2,\ldots]\colon \sum_{i=1}^ka_i=n\text{ for some \(k\in\mathbb N\)}\}.NEWLINE\]NEWLINE The following results are proved using Farey maps and the infinite ergodic theorem: NEWLINENEWLINENEWLINEDenote by \(\lambda\) the Lebesgue measure on \([0,1]\). Then NEWLINENEWLINE\[NEWLINE\begin{aligned} \lim_{n\to\infty}\lambda(\mathcal C_n)&=0,\\ \sum_{k=1}^n\lambda(\mathcal C_k)&\sim{n\over\log_2n},\\ \lambda(\mathcal C_n) &\sim{1\over\log_2n}.\end{aligned}NEWLINE\]NEWLINE The authors give two proofs for the first statement.