The M\"{o}bius transformation of continued fractions with bounded upper and lower partial quotients

TL;DR

This paper extends existing results by establishing bounds on the continued fraction expansion of a Möbius transformation of a real number, based on the bounds of the original number's continued fraction, with implications for understanding transformations with bounded partial quotients.

Contribution

It provides a new bound for the continued fraction of transformed numbers under Möbius transformations with integer entries, extending prior work by Stambul.

Findings

Bound on continued fractions of h(x) derived from bounds of x

Extension of Stambul's result to more general Möbius transformations

Implications for numbers with bounded partial quotients

Abstract

Let $h$: $x\mapsto \frac{ax+b}{cx+d} $ be the nondegenerate M\"{o}bius transformation with integer entries. We get a bound of the continued fraction of $h(x)$ by the upper and lower bound of continued fraction of $x$, which extends a result of Stambul \cite{S2}.