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
This study is the complex analogue of a well-known result of Wang and Wu [Adv. Math. 218 (2008), no. 5, 1319–1339].