The relative growth rate for partial quotients

The rate of growth of the partial quotients of an irrational number is studied relative to the rate of approximation of the number by its convergents. The focus is on the Hausdorff dimension of exceptional sets on which different growth rates are achieved. In this note we look at the rate of growth of the partial quotients ai of the irrational number x = [a1, a2, . . . ] = 1 a1 + 1 a2 + 1 a3 + · · · relative to the rate at which x is approximated by its rational convergents. For x ∈ (0, 1) irrational, let {n qn } be the sequence of rational convergents given by the continued fraction expansion of x [7]. It follows from classical results of Khinchin and Lévy [3, 5] that for almost all x (1) lim n→∞ log an n = 0 and lim n→∞− log |x− pn qn | n = π2 6 log 2 . Consequently, for almost all x ∈ (0, 1) (2) lim n→∞ log an+1 log |x− pn qn | = 0. Here we study the Hausdorff dimension of exceptional sets on which the limit (2) either does not exist or is different from zero. Similar, nonoverlapping, problems are considered in [9] using more sophisticated methods of multifractal analysis. The original motivation for looking at this question Received November 12, 2007. Mathematics Subject Classification. 11K50, 11K60.