On a theorem of Serret on continued fractions

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 $\text {PGL}(2,\mathbb {Z})$ 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.