A Central Limit Theorem for Rosen Continued Fractions

TL;DR

This paper establishes a central limit theorem for Rosen continued fractions by analyzing the spectral properties of the associated transfer operator, providing a rigorous probabilistic limit law for these number-theoretic transformations.

Contribution

It introduces a new proof of the spectral gap for Rosen continued fractions, extending the CLT to a broader class of continued fraction algorithms.

Findings

Proves a CLT for Rosen continued fractions.

Establishes a Lasota-Yorke bound for general continued fractions.

Demonstrates the spectral gap and quasi-compactness of the transfer operator.

Abstract

We prove a central limit theorem for Birkhoff sums of the Rosen continued fraction algorithm. A Lasota-Yorke bound is obtained for general one-dimensional continued fractions with the bounded variation space, which implies quasi-compactness of the transfer operator. The main result is a direct proof of the existence of a spectral gap, assuming a certain behavior of the transformation when iterated. This condition is explicitly proved for the Rosen system. We conclude via well-known results of A. Broise that the central limit theorem holds.