On convergence rate in the Gauss-Kuzmin problem for $\theta$-expansions

TL;DR

This paper investigates the convergence rate of $ heta$-expansions by establishing a two-dimensional Gauss-Kuzmin theorem and providing explicit bounds on the error term, enhancing understanding of their dynamical properties.

Contribution

It introduces a two-dimensional Gauss-Kuzmin theorem for $ heta$-expansions and derives explicit bounds on the convergence error, advancing the analysis of these expansions.

Findings

Proved a two-dimensional Gauss-Kuzmin theorem for $ heta$-expansions.

Derived explicit lower and upper bounds for the convergence error.

Provided numerical estimates for the convergence rate.

Abstract

After providing an overview of $\theta$-expansions introduced by Chakraborty and Rao, we focus on the Gauss-Kuzmin problem for this new transformation. Actually, we complete our study on these expansions by proving a two-dimensional Gauss-Kuzmin theorem. More exactly, we obtain such a theorem related to the natural extension of the associated measure-dynamical system. Finally, we derive explicit lower and upper bounds of the error term which provide interesting numerical calculations for the convergence rate involved.