Levy-Khintchin Theorem for best simultaneous Diophantine approximations

Yitwah Cheung Nicolas Chevallier

Dynamical Systems Number Theory mathscidoc:2207.11001

Annales Scientifiques de l'Ecole Normale Superieure
We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products q_n d(q_n θ,Z) where the qn's are the denominators of the convergents associated with the real number θ by the ordinary continued fraction algorithm. Beside these two main results, we show that when d≥2, for almost all vectors θ∈R^d, lim inf_{n→∞} q_{n+k} d(q_n θ,Z^d)=0 for all positive integers k, where (q_n)_{n∈N} is the sequence of best approximation denominators of θ.
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@inproceedings{yitwahlevy-khintchin,
  title={Levy-Khintchin Theorem for best simultaneous Diophantine approximations},
  author={Yitwah Cheung, and Nicolas Chevallier},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220702152259824515523},
  booktitle={Annales Scientifiques de l'Ecole Normale Superieure},
}
Yitwah Cheung, and Nicolas Chevallier. Levy-Khintchin Theorem for best simultaneous Diophantine approximations. In Annales Scientifiques de l'Ecole Normale Superieure. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220702152259824515523.
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