Renewal-type Limit Theorem for Continued Fractions with Even Partial Quotients

TL;DR

This paper establishes a renewal-type limit theorem for continued fractions with even partial quotients, demonstrating the existence of a limiting distribution for their denominators, extending classical results to a new class of continued fractions.

Contribution

It introduces a renewal-type limit theorem for continued fractions with even partial quotients, using a novel approach based on natural extensions and mixing properties.

Findings

Existence of a limiting distribution for denominators

Construction of a natural extension of a Gauss-like map

Proof of mixing for a related special flow

Abstract

We prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by F. Schweiger and studied also by C. Kraaikamp and A. Lopes. Our main result is proven following the strategy used by Ya. Sinai and C. Ulcigrai in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.