Subexponentially increasing sums of partial quotients in continued fraction expansions

Abstract

Abstract We investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S_{n}(x)=\sum_{j=1}^n a_{j}(x)$ , where x = [ a 1 ( x ), a 2 ( x ), . . .] is the continued fraction expansion of an irrational x ∈ (0, 1). Precisely, for an increasing function ϕ : $\mathbb{N}$ → $\mathbb{N}$ , one is interested in the Hausdorff dimension of the set E_\varphi = \left\{x\in (0,1): \lim_{n\to\infty} \frac {S_n(x)} {\varphi(n)} =1\right\}. Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case exp( n γ ), γ ∈ [1/2, 1). We show that when γ ∈ [1/2, 1), E ϕ has Hausdorff dimension 1/2. Thus, surprisingly, the dimension has a jump from 1 to 1/2 at ϕ( n ) = exp( n 1/2 ). In a similar way, the distribution of the largest partial quotient is also studied.