On a theorem of Serret on continued fractions

TL;DR

This paper provides an upper bound on the indices at which two irrational numbers related by a PGL(2,Z) transformation have coinciding continued fraction quotients, extending Serret's classical theorem.

Contribution

It introduces a quantitative bound on the indices in Serret's theorem relating irrational numbers via PGL(2,Z) transformations.

Findings

Derived an explicit upper bound for indices s and t

Extended classical Serret theorem with quantitative estimates

Applicable to understanding the structure of continued fractions under transformations

Abstract

A classical theorem in continued fractions due to Serret shows that for any two irrational numbers x and y related by a transformation $\gamma$ in PGL(2,Z) there exist s and t for which the complete quotients x_s and y_t coincide. In this paper we give an upper bound in terms of $\gamma$ for the smallest indices s and t.