On a Gauss-Kuzmin-Type Problem for a Family of Continued Fraction Expansions

TL;DR

This paper investigates a specific family of continued fraction expansions with digits derived from powers of an integer, analyzing their invariant measures and solving a Gauss-Kuzmin type problem using advanced probabilistic methods.

Contribution

It introduces a new family of continued fraction expansions and applies the method of random systems with complete connections to solve the Gauss-Kuzmin problem for this family.

Findings

Derived the invariant measure for the expansion transformation

Solved the Gauss-Kuzmin type problem for the new family

Analyzed the Perron-Frobenius operator associated with the expansion

Abstract

In this paper we study in detail 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 the transformation which generates this expansion and its invariant measure, the Perron-Frobenius operator is given and studied. For this expansion, we apply the method of random systems with complete connections by Iosifescu and obtained the solution of its Gauss-Kuzmin type problem.