Some asymptotic results on extended sequences connected with the regular continued fraction (Q2778500)

In a regular continued fraction expansion of irrational numbers \(x\), let \(a_j\) be the digits of \(x\), and let \(\mu\) be any measure on the real line that assigns zero measure to rational numbers. Define the sequences \(r_n=1/ \tau^{n-1}\), where \(\tau\) is the continued fraction transformation operator, and \(s_n=1/ (a_n+s^{n-1})\). Upon extending \(\mu\) to two dimensions, the author uses Wirsing's method developed for the one-dimensional case for solving Gauss' problem and some asymptotic independence of shifts of the sequences \(r_n\) and \(s_n\) is established.