The bisequence of approximation coefficients for Gauss-like and Renyi-like maps on the interval

TL;DR

This paper investigates the properties of bi-sequences of approximation coefficients linked to Gauss-like and Renyi-like maps, generalizing classical Diophantine approximation results for these continued fraction transformations.

Contribution

It establishes new arithmetic and geometric properties of these bi-sequences for families of Mobius transformations, extending known theorems in Diophantine approximation.

Findings

Derived properties of bi-sequences for Gauss and Renyi maps

Generalized classical Diophantine approximation theorems

Connected the properties to regular and backwards continued fractions

Abstract

We will establish several arithmetic and geometric properties regarding the bi-sequences of approximation coefficients (BAC) associated with the two one-parameter families of piecewise-continuous Mobius transformations introduced by Haas and Molnar. The Gauss and Renyi maps, which lead to the expansions of irrational numbers on the interval as regular and backwards continued fractions, are realized as special cases. The results are natural generalizations of theorems from Diophantine approximation.