Levy-Khintchin Theorem for best simultaneous Diophantine approximations

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 $q_n d(q_nθ, Z)$ where the $q_n$'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 $d\ge2$, for almost all vectors $θ\in R^d$, $\liminf_{n\to\infty} q_{n+k}d(q_nθ, Z^d)=0$ for all positive integers $k$, where $(q_n)_{n\in N}$ is the sequence of best approximation denominators of $θ$.

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