Renewal-type Limit Theorem for the Gauss Map and Continued Fractions

TL;DR

This paper proves a renewal-type limit theorem for continued fractions, showing that the ratio of the denominator of convergents to a threshold R converges in distribution as R grows large, using mixing properties of a special flow.

Contribution

It establishes the existence of a limiting distribution for the ratio of denominators of continued fraction convergents to a threshold, extending renewal theory to the Gauss map context.

Findings

The ratio q_{n_R}/R converges in distribution as R approaches infinity.

The proof utilizes mixing properties of a special flow over the Gauss map.

The result provides a new understanding of the asymptotic behavior of continued fraction convergents.

Abstract

In this paper we prove the following renewal-type limit theorem. Given an irrational $\alpha$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of the convergents of $\alpha$ which exceeds R. The main result in the paper is that the ratio $q_{n_R}/R$ has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map.