Viviane Baladi and Aicha Hachemi
A local limit theorem with speed of convergence 
for Euclidean algorithms and diophantine costs
(78K, AMS LaTeX)

ABSTRACT.  For large N, we consider the ordinary continued fraction 
of x=p/q with 1<= p <= q<= N, or, equivalently, Euclid's 
gcd algorithm for two integers 1<= p <= q\le N, 
putting the uniform distribution on the set of p and q's. 
We study the distribution of the total cost of execution 
of the algorithm for an additive cost function 
c on the set Z_+ of possible digits, 
asymptotically for N going to infinity. 
If c is nonlattice 
and satisfies mild growth conditions, the local limit theorem was 
proved previously by the second named author. 
Introducing diophantine conditions on the cost, we are 
able to control the speed of convergence in the local limit 
theorem, by using previous estimates of the first author 
and Valleee, as well as bounds of Dolgopyat and Melbourne on 
transfer operators.