The Hurwitz continued fraction expansion as applied to real numbers

TL;DR

This paper explores the relationship between Hurwitz's complex continued fraction algorithm and the classical real continued fraction algorithm, clarifying their connection and revalidating a key result in the field.

Contribution

It clarifies how Hurwitz's complex continued fraction algorithm relates to the classical real algorithm and re-proves a significant existing result.

Findings

Established the connection between Hurwitz's and classical continued fractions on reals

Reproved the main result of Choudhuri and Dani (2015)

Enhanced understanding of Hurwitz's algorithm in the real case

Abstract

Hurwitz (1887) defined a continued fraction algorithm for complex numbers which is better behaved in many respects than a more "natural" extension of the classical continued fraction algorithm to the complex plane would be. Although the Hurwitz complex continued fraction algorithm is not "reducible" to another complex continued fraction algorithm, over the reals the story is different. In this note we make clear the relation between the restriction of Hurwitz's algorithm to the real numbers and the classical continued fraction algorithm. As an application we reprove the main result of Choudhuri and Dani (2015).