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On sums of partial quotients in Hurwitz continued fraction expansions

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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 \(0<C\le \alpha_1\le \alpha_2<\infty \).

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Acknowledgement

We thank the referee for careful reading of the paper and helpful comments and corrections which improved the paper substantially.

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Correspondence to Q. Xiao.

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He, Y., Xiao, Q. On sums of partial quotients in Hurwitz continued fraction expansions. Acta Math. Hungar. 170, 17–32 (2023). https://doi.org/10.1007/s10474-023-01345-3

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  • DOI: https://doi.org/10.1007/s10474-023-01345-3

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