On the distribution for sums of partial quotients in continued fraction expansions

Let x is an element of [0, 1) and [a(1)(x), a(2)(x),...] be the continued fraction expansion of x. For any n >= 1, write S-n(x) = Sigma(n)(k=1) a(k)(x). Khintchine (1935 Compos. Math. 1 361-82) proved that S-n(x)/n log n converges in measure to 1/log 2 with respect to L-1, where L-1 denotes the one dimensional Lebesgue measure. Philipp (1988 Monatsh. Math. 105 195-206) showed that there is not a reasonable normalizing sequence such that a strong law of large numbers is satisfied. In this paper, we show that for any alpha >= 0, the set E(alpha) = {x is an element of [0, 1): lim(n -> 8) S-n(x)/n log n = alpha} is of Hausdorff dimension 1. Furthermore, we prove that the Hausdorff dimension of the set consisting of reals whose sums of partial quotients grow at a given polynomial rate is 1.