We improve a result of D. Knuth about the convergence of approximations of a continued fraction.
1998 Academic Press 0. INTRODUCTION Recently several authors have been interested by the convergence of distribution functions for various quantities related to the continued fraction expansion and more precisely by an analogue of the Gauss Kuzmin problem. In probabilistic terms, let x be a randomly choosen point in [0, 1]. We denote by x=[0; a 1 , a 2 , ...] its continued fraction expansion and as usual by p n รq n =[0; a 1 , ..., a n ] the n th convergent. Let (x n ) n 1 be a sequence of random variables and related (in some sense) to the continued fraction expansion of x. We ask if there exist positive constants C, r with 0<r<1 and a distribution function h such that
for all n 1. The first and the most celebrated result in this direction is the Gauss Kuzmin problem (which has a long history since the letter of Gauss to Laplace in 1812) where x n is taken to be T n x where T:
is the Gauss transtormation of continued fractions defined by Tx=(1รx) (mod 1). In this case, we know now that (1) holds with h(z)=(log 2) &1 log(1+z), for 0 z 1, and the best value for r is : & 0.303663 with six exact digits. This value has been determined for the first time by Wirsing [18] after several improvements by Kuzmin [13], Le vy [14], and Szu sz [17]. The Article No. NT972186