On sums of partial quotients in continued fraction expansions

Abstract

Assume x ∊ [0, 1) taking on its continued fraction expansion as [a1(x), a2(x), …]. For any n ⩾ 1, write . Khintchine (1935 Compos. Math. 1 361–82) proved that Sn(x)/(n log n) converges in measure to 1/log 2 with respect to , where denotes the one-dimensional Lebesgue measure. Philipp (1988 Monatsh. Math. 105 195–206) showed that {an(x), n ⩾ 1} cannot satisfy a strong law of large numbers for any reasonably growing norming sequence. In (Wu and Xu 2008 Preprint), we discussed the sets of continued fractions whose sums of partial quotients tend to infinity with the polynomial growth rate. In this paper, we consider the sets of continued fractions whose sums of partial quotients tend to infinity exponentially and doubly exponentially. The Hausdorff dimensions of such sets are determined.