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Gaussian behavior of quadratic irrationals

Volume 197 / 2021

Eda Cesaratto, Brigitte Vallée Acta Arithmetica 197 (2021), 159-205 MSC: 11K50, 11M36, 37D20, 60F05. DOI: 10.4064/aa191205-18-5 Published online: 5 October 2020

Abstract

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.

Authors

  • Eda CesarattoInstituto del Desarrollo Humano
    Universidad Nacional de Gral. Sarmiento
    and
    National Council of Science and Technology (CONICET)
    J. M. Gutiérrez 1150
    (B1613GSX) Los Polvorines, Buenos Aires, Argentina
    e-mail
  • Brigitte ValléeGREYC, Université de Caen and CNRS
    Bâtiment Sciences 3, Université de Caen
    Bd Maréchal Juin
    F-14032 Caen Cedex, France
    e-mail

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