Gaussian behaviour of quadratic irrationals
Résumé
We study the probabilistic behavior of the continued fraction expansion of a quadratic irrational number, when weighted by some “additive” cost. We prove asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal with the underlying dynamical system associated with the Gauss map, and its weighted periodic trajectories. We work with analytic combinatorics methods, and mainly bivariate Dirichlet generating functions; we use various tools, from number theory (the Landau Theorem), probability (the Quasi-Powers Theorem), or dynamical systems: our main object of study is the (weighted) transfer operator, which we relate to the generating functions of interest. The present paper exhibits strong parallelism between periodic trajectories and rational trajectories. We indeed extend the general framework which has been previously described by Baladi and Vallée for rational trajectories. However, our extension to quadratic irrationals needs deeper functional analysis properties.