Levy-Khintchin Theorem for best simultaneous Diophantine approximations

Abstract

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 $q_n d(q_n\theta, Z)$ where the $q_n$'s are the denominators of the convergents associated with the real number $\theta$ by the ordinary continued fraction algorithm. Beside these two main results, we show that when $d\ge2$, for almost all vectors $\theta\in R^d$, $\liminf_{n\to\infty} q_{n+k}d(q_n\theta, Z^d)=0$ for all positive integers $k$, where $(q_n)_{n\in N}$ is the sequence of best approximation denominators of $\theta$.