Yann Bugeaud, Gerardo Gonzalez Robert and I have uploaded a paper to the arXiv https://arxiv.org/abs/2306.08254 concerning the metrical properties of Hurwitz Continued Fraction expansions.
We develop the geometry of Hurwitz continued fractions – a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. We obtain a detailed description of the shift space associated with Hurwitz continued fractions and, as a consequence, we contribute significantly in establishing the metrical theory of Hurwitz continued fractions, analogous to the well-established theory of regular continued fractions for real numbers.
Let 
be any function and 
denote the 
th partial quotient in the Hurwitz continued fraction of a complex number 
. The main result of the paper is the Hausdorff dimension analysis of the set
![\[E(\Phi):= \left\{ z\in \mathbb C: |a_n(z)|\geq \Phi(n) \text{for infinitely many }n\in\mathbb{N} \right\}.\]](https://drmumtazhussain.com/wp-content/ql-cache/quicklatex.com-9b841b694db2e19c8dda4574dc993662_l3.png "Rendered by QuickLaTeX.com")
This study is the complex analogue of a well-known result of Wang and Wu [Adv. Math. 218 (2008), no. 5, 1319–1339].