Metrical properties of Hurwitz continued fractions

We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued fractions, among other things, we determine that the space of valid sequences is not a closed set of sequences. Additionally, we establish a comprehensive metrical theory for Hurwitz continued fractions. Let phi : N -> R->0 be any function. For any complex number z and n is an element of N, let a(n)(z) denote the nth partial quotient in the Hurwitz continued fraction of z. One of the main results of this paper is the computation of the Hausdorff dimension of the set E(phi) := {z is an element of C : |a(n)(z)| >= phi(n) for infinitely many n is an element of N}. This study is a complex analog of a well-known result of Wang and Wu (2008) [55]. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).