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 Sn(x) = ∑n j=1 aj(x), where x = [a1(x), a2(x), · · · ] is the continued fraction expansion of an irrational x ∈ (0, 1). Precisely, for an increasing function φ : N → N, one is interested in the Hausdorff dimension of the sets Eφ = { x ∈ (0, 1) : lim n→∞ Sn(x) φ(n) = 1 } . Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case exp(n), γ ∈ [1/2, 1). We show that when γ ∈ [1/2, 1), Eφ has Hausdorff dimension 1/2. Thus, surprisingly, the dimension has a jump from 1 to 1/2 at φ(n) = exp(n). In a similar way, the distribution of the largest partial quotient is also studied.