06-119 Viviane Baladi and Aicha Hachemi

A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs (78K, AMS LaTeX) Apr 16, 06

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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.

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