Intermediate convergents and a metric theorem of Khinchin

TL;DR

This paper extends Khinchin's metric theorem for continued fractions by examining intermediate convergents with denominators below a threshold, providing a quantitative asymptotic result using classical and modern methods.

Contribution

It introduces a reformulation of Khinchin's theorem focusing on intermediate convergents and derives a new quantitative asymptotic theorem in this context.

Findings

Established a limit theorem for sums over intermediate convergents

Provided asymptotic estimates analogous to classical results

Combined classical ideas with modern techniques for new insights

Abstract

A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function $f$ defined on the positive integers and a real number $x$, and form the partial sums $s_n$ of $f$ evaluated at the partial quotients $a_1,..., a_n$ in the continued fraction expansion for $x$. Does the sequence $\{s_n/n\}$ have a limit as $n\rar\infty$? In 1935 A. Y. Khinchin proved that the answer is yes for almost every $x$, provided that the function $f$ does not grow too quickly. In this paper we are going to explore a natural reformulation of this problem in which the function $f$ is defined on the rationals and the partial sums in question are over the intermediate convergents to $x$ with denominators less than a prescribed amount. By using some of Khinchin's ideas together with more modern results we are able to provide a quantitative asymptotic theorem analogous to the classical one mentioned above.