Geometric and Diophantine aspects of Complex Continued Fractions

Wednesday, 31 May 2023

Abstract

Regular continued fractions have been a useful tool for over a century. In this talk we will discuss the develop geometry of Hurwitz continued fractions-- a major tool in understanding the approximation properties of complex numbers by ratios of Gausian integers. In doing so, we completely determine the Hausdorff dimension of sets of points with their partial quotients in the Hurwitz continued fractions grows at a certain rate. Let $\Phi$ be a positive function then the Hausdorff dimension of the set

\[E(\Phi) \colon=\left\{ z\in \mathbb C: |a_n(z)|\geq \Phi(n) \text{ for infinitely many }n\in\mathbb{N} \right\}\]

is fully characterized. As a consequence of our work, we obtained a detailed description of the shift space associated to Hurwitz continued fractions as well as a complex version of T. {\L}uczak's on sets of continued fractions with rapidly growing partial quotients. Our results contribute significantly in developing the metrical theory of Hurwitz continued fractions akin to regular continued fractions for real numbers.

This is a joint work with Mumtaz Hussain and Yann Bugeaud.