Application of the subadditive ergodic theorem to the metric theory of continuous fractions

For any irrational x is an element of [0, 1] We denote by p(n)(x)/q(n)(x), n = 1, 2,... the sequence of its continued fraction convergents and define 0(n)(x) := q(n) \q(n)x-p(n)\. Also let T: [0, 1] --> [0, 1] be defined by T(0) = 0 and T(x) = 1/x - [1/x] if x not equal 0. For some random variables X-1,X-2,..., which are connected with the regular continued fraction expansion, the subadditive ergodic theorem yields to the existence of a Function omega satisfying: for all z is an element of R, lim(n-->+infinity)1/n#{1 less than or equal to i less than or equal to n/X-i(z)less than or equal to z} = omega(z) for almost every x. In particular, for X-n = theta(n), using this study and a result of Knuth, we give another proof of the following conjecture of Lenstra (the first proof of this conjecture has been given by Bosma, Jager, and Wiedijk): for all z is an element of [0, 1], [GRAPHICS] for almost every x. Furthermore, for X-n = 0(n) circle T-n and X-n = (q(n-1)/q(n)) circle T-n, the functions omega are explicitly determined. The above results show that the subadditive ergodic theorem can be useful in the metric theory of continued fraction. (C) 1997 Academic Press.