Dr Mumtaz Hussain


Single post

Metrical properties of Hurwitz Continued Fractions

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 \Phi:\mathbb N\to \mathbb R_{\geq 0} be any function and a_n(z) denote the nth partial quotient in the Hurwitz continued fraction of a complex number z. The main result of the paper is the Hausdorff dimension analysis of the set

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

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

Write a Comment

Your email address will not be published. Required fields are marked *