IJPAM: Volume 2, No. 3 (2002)

LINEAR FRACTIONAL TRANSFORMATION OF
CONTINUED FRACTIONS WITH BOUNDED
PARTIAL QUOTIENTS: ARITHMETICAL
AND GEOMETRICAL POINT OF VIEW

A.K. Ben-Naoum
Centre d'Ingénierie des Systèmes
d'Automatique et de Mécanique Appliquée
Faculté des Sciences Appliquées
Université Catholique de Louvain
Bâtimeut Euler, 4-6, avenue Georges Lemaître
B-1348 Louvain-la-Neuve, BELGIUM
[email protected]


Abstract.In this paper I will give new proofs of some simple theorems concerning continued fractions. Like J.O. Shallit said, ``the proofs in the literature seem to be missing, incomplete, or hard to locate". In paticular, I will give two proofs of the following ``folk theorem": if $\alpha$ is an irrational number whose continued fraction has bounded partial quotients, then any non-trivial linear fractional transformation of $\alpha$ also has bounded partial quotients. The first proof is based upon arithmetics arguments and the second one upon the geometrical interpretation of the best approximations to $\alpha$. The result is a consequence of the following inequality due to Lagarias and Shallit [#!LS!#]:

\begin{displaymath}\frac{1}{\vert ad-bc\vert}L(\alpha) \leq L\left(\frac{a\alpha+b}{c\alpha+d}\right)\leq\vert ad-bc\vert L(\alpha)\,,\end{displaymath}

where $a, b,c, d \; \in {\mathbb Z},$ with $\vert ad-bc\vert\not=0$ and $L(\alpha)$ is the Lagrange constant of $\alpha$.

Received: March 7, 2002

AMS Subject Classification: 11J04, 11J70

Key Words and Phrases: Lagrange constant, continued fractions, best approximations

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2002
Volume: 2
Issue: 3