A three-dimensional probability distribution in the metrical theory of continued fractions

Let an, n ≥ 1, denote the incomplete quotients of the continued fraction expansion of an arbitrary irrational number in the unit interval I = [0, 1]. For any a ∈ I put s n+1 = 1/(an+1+ s a n ), u n+1 = s a n + 1/τ n, n ≥ 0, with s 0 = a, where τ is the continued fraction transformation, and let γa be the probability measure on the Borel subsets of I defined by its distribution function γa([0, x]) = (a + 1)x ax + 1 , x ∈ I. We study the joint distribution function of sa n , τ n, and u n+1, n ≥ 0, under γa , a ∈ I . We derive the asymptotic distribution function, lower and upper bounds for the error as well as its optimal convergence rate to 0 as n→∞. The same problems are taken up for the distributions of the pairs (τn, u n+1) and (s a n , u a n+1) under γa , a ∈ I .