A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs

Details A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs

2006-04-14

arxiv.org/abs/math/0604341v3

Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques 2008, Vol. 44, No. 4, 749-770

Mathematics

Dynamical Systems

Published in at http://dx.doi.org/10.1214/07-AIHP140 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiqu

Scientific paper

10.1214/07-AIHP140

For large $N$, we consider the ordinary continued fraction of $x=p/q$ with $1\le p\le q\le N$, or, equivalently, Euclid's gcd algorithm for two integers $1\le p\le q\le N$, 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 $\mathbb{Z}_+^*$ of possible digits, asymptotically for $N\to\infty$. 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. We use previous estimates of the first author and Vall\'{e}e, and we adapt to our setting bounds of Dolgopyat and Melbourne on transfer operators. Our diophantine condition is generic (with respect to Lebesgue measure). For smooth enough observables (depending on the diophantine condition) we attain the optimal speed.

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