Subexponentially increasing sums of partial quotients in continued fraction expansions

We investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients S-n(x) = Sigma(n)(j=1) a(j) (x), where x = [a(1)(x), a(2)(x),...] is the continued fraction expansion of an irrational x is an element of (0, 1). Precisely, for an increasing function phi : N -> N, one is interested in the Hausdorff dimension of the set E-phi = { x is an element of (0, 1) : lim(n -> 8) S-n(x)/phi(n) = 1}. Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case exp(n gamma), gamma is an element of [1/2, 1). We show that when gamma is an element of [1/2, 1), E-phi has Hausdorff dimension 1/2. Thus, surprisingly, the dimension has a jump from 1 to 1/2 at phi(n) = exp(n(1/2)). In a similar way, the distribution of the largest partial quotient is also studied.