On sums of partial quotients in Hurwitz continued fraction expansions

Abstract

Assume $z\in \mathfrak F:=\{x+iy: x,y\in[-1/2,1/2)\}$ has its Hurwitz continued fraction expansion $[0;a_1(z),a_2(z),\dots]$ where $a_j(z)$ are Gaussian integers and $|a_j(z)|\ge \sqrt 2$. For any $n\ge 1$, write $S_n(z)=\sum_{j=1}^{n}a_j(z)$ and $R_n(z)=\sum_{j=1}^{n}|a_j(z)|$ . It is known that for $\mathcal L^2$-almost every $z$ , $S_n(z)/n$ and $R_n(z)/n$ converge to the constants $W$ and $C$ respectively, where $\mathcal L^2$ denotes the 2-dimensional Lebesgue measure. We show that the sets

$$E(w_1,w_2):=\bigl\{z\in \mathfrak F:\text{the accumulation points of }\bigl(\frac{S_n(z)}{n}\bigr)_{n\ge 1}\text{ are } w_1\text{ and }w_2\bigr\}$$

and

$$F(\alpha_1,\alpha_2):=\bigl\{z\in \mathfrak F:\liminf_{n\to\infty}\frac{R_n(z)}{n}=\alpha_1\text{ and } \limsup_{n\to\infty}\frac{R_n(z)}{n}=\alpha_2\bigr\}$$

have full Hausdorff dimensions, where $w_1, w_2\in \mathbb{C}$ and $0<C\le \alpha_1\le \alpha_2<\infty$.