On the Gauss-Kuzmin-L\'evy problem for nearest integer continued fractions

TL;DR

This paper establishes effective bounds for the Gauss-Kuzmin-Lévy problem related to nearest integer continued fractions, providing asymptotic formulas with explicit error terms for certain Gauss-type transformations.

Contribution

It offers explicit bounds and asymptotic formulas for the distribution of iterates of specific continued fraction transformations, improving understanding of their statistical properties.

Findings

Asymptotic formulas with error term q^n where q=0.288

Effective bounds for the Gauss-Kuzmin-Lévy problem

Comparison with Wirsing constant q_W=0.3036

Abstract

This note provides an effective bound in the Gauss-Kuzmin-L\'evy problem for some Gauss type shifts associated with nearest integer continued fractions, acting on the interval $I_0=[0,\frac{1}{2}]$ or $I_0=[-\frac{1}{2},\frac{1}{2}]$. We prove asymptotic formulas $\lambda (T^{-n}I) =\mu(I)(\vert I_0 \vert +O(q^n))$ for such transformations $T$, where $\lambda$ is the Lebesgue measure on $\mathbb R$, $\mu$ the normalized $T$-invariant Lebesgue absolutely continuous measure, $I$ subinterval in $I_0$, and $q=0.288$ is smaller than the Wirsing constant $q_W=0.3036\ldots$