Continued fraction expansions for complex numbers - a general approach

TL;DR

This paper develops a comprehensive framework for continued fraction expansions of complex numbers, proving convergence, an analogue of Lagrange's theorem, and properties like monotonicity and exponential growth for specific algorithms.

Contribution

It introduces a general approach to complex continued fractions, establishes convergence results, and extends classical theorems to complex settings with new algebraic and growth properties.

Findings

Proved convergence of complex continued fraction sequences.

Established an analogue of Lagrange's theorem for quadratic surds in complex numbers.

Demonstrated exponential growth of denominators for Eisenstein integer algorithms.

Abstract

We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial quotients in a discrete subring of $\mathbb C$ an analogue of the classical Lagrange theorem, characterising quadratic surds as numbers with eventually periodic continued fraction expansions, is proved. Monotonicity and exponential growth are established for the absolute values of the denominators of the convergents for a class of continued fraction algorithms with partial quotients in the ring of Eisenstein integers.