Arithmetic diophantine approximation for continued fractions-like maps on the interval

TL;DR

This paper explores the arithmetical properties and bounds of approximation coefficients in continued fraction-like expansions generated by Möbius transformation-based maps, extending classical continued fraction theory.

Contribution

It introduces a new class of continued fraction-like maps derived from Möbius transformations and analyzes their arithmetical properties and approximation bounds.

Findings

Established bounds for approximation coefficients

Extended classical continued fraction theory

Analyzed properties of new continued fraction-like maps

Abstract

We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the fractional part of M$\operatorname{\ddot{o}}$bius transformations which carry the end points of the unit interval to zero and infinity, extending the classical regular and backwards continued fractions expansions.