A Gauss-Kuzmin Theorem for Some Continued Fraction Expansions

TL;DR

This paper extends the Gauss-Kuzmin theorem to a new family of continued fraction expansions based on differences of powers, providing a rigorous solution to their distributional properties.

Contribution

It introduces a novel class of continued fraction expansions and applies existing methods to establish their Gauss-Kuzmin type theorem.

Findings

Established the distributional behavior of the new continued fraction expansions.

Provided a rigorous proof of the Gauss-Kuzmin type theorem for these expansions.

Abstract

We consider a family of continued fraction expansions of any number in the unit closed interval $[0,1]$ whose digits are differences of consecutive non-positive integer powers of an integer $m \geq 2$. For this expansion, we apply the method of Rockett and Sz\"usz from [6] and obtained the solution of its Gauss-Kuzmin type problem.