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 result is the Levy-Khintchin theorem about the almost sure growth rate of the denominators of the convergents. The second result is a theorem of Doeblin and of Bosma, Jager 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. Besides these two main results, we show that when d >= 2, for almost all vectors theta is an element of R-d, lim inf(n ->infinity) q(n+k) d(q(n) theta, Z(d))(d) = 0 for all positive integers k, where (q(n))(n is an element of N) is the sequence of best approximation denominators of theta.