Euclidean algorithms are Gaussian

Abstract

This study provides new results about the probabilistic behaviour of a class of Euclidean algorithms: the asymptotic distribution of a whole class of cost-parameters associated to these algorithms is normal. For the cost corresponding to the number of steps Hensley already has proved a Local Limit Theorem; we give a new proof, and extend his result to other euclidean algorithms and to a large class of digit costs, obtaining a faster, optimal, rate of convergence. The paper is based on the dynamical systems methodology, and the main tool is the transfer operator. In particular, we use recent results of Dolgopyat.