The Khintchine constants for generalized continued fractions

Abstract

Let T-p(x) - 1/x(p) (mod 1) for 0 < x < 1 and T-p(0) = 0. It is known that if p > p(0) = 0.241485.... then there exists an ergodic invariant measure of the form rho(p)dx. Let a(n) = [(1/T-p(n-1)(x))(p)], n greater than or equal to 1, where [t] is the integer part of t. If p = 1, then a(1), a(2),...,a(n) are the partial quotients of the classical continued fraction of x. For a real number q, we consider averages of a(n): [GRAPHICS] We show that (i) for almost every x, K-p,K-q : = lim(n-->infinity)K (p, q, n, x) < ∞ if and only if q < 1/p, (ii) lim(p-->infinity) (log K-p,K-q)/p = 1 if q = 0 where log denotes the natural logarithm, (iii) lim(p-->infinity) log K-p,K-q/ log p = 1 /\q\ if q < 0. The limiting behavior of K-p,K-q is investigated as p ↓ p(o) with computer simulations. (C) 2002 Elsevier Inc. All rights reserved.