Limit theorems for counting large continued fraction digits

TL;DR

This paper proves a central limit theorem for counting large digits in continued fractions, refining previous results and providing new insights into digit distribution and mixing properties of the Gauss system.

Contribution

It introduces a central limit theorem for large continued fraction digits with sequences tending to infinity and refines the Borel-Bernstein theorem for digit occurrence intervals.

Findings

Established a CLT for counting large continued fraction digits.

Refined the Borel-Bernstein theorem for digit intervals.

Explicitly determined the first $6$-mixing coefficient for the Gauss system.

Abstract

We establish a central limit theorem for counting large continued fraction digits $(a_n)$, i.e. we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp which additionally assumes that $b_n$ tends to infinity. Moreover, we give a refinement of the famous Borel-Bernstein Theorem for continued fractions regarding the event that the $n$-th continued fraction digit lies infinitely often between $d_n$ and $d_n(1+1/c_n)$ for given sequences $(c_n)$ and $(d_n)$. Also for these sets we obtain a central limit theorem. As an interesting side result we determine the first $\phi$-mixing coefficient for the Gauss system explicitly.