Generalized continued fraction expansions of complex numbers, and applications to quadratic and badly approximable numbers

TL;DR

This paper explores generalized continued fraction expansions of complex numbers within Euclidean subrings, focusing on Gaussian and Eisenstein integers, and applies group actions to analyze quadratic surds and badly approximable numbers, extending classical approximation properties.

Contribution

It introduces a unified framework for continued fractions of complex numbers, including zeroes of binary forms, and generalizes best approximation properties to broader classes.

Findings

Extended continued fraction theory to complex numbers in Euclidean rings.

Applied group actions to analyze quadratic surds and badly approximable numbers.

Generalized best approximation properties for complex continued fractions.

Abstract

We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common approach to studying the continued fraction expansions of zeroes of binary forms, via consideration of the action of the general linear group, and apply it to discuss expansions of quadratic surds on the one hand and of badly approximable numbers on the other hand. Also, we generalize the property of the simple continued fraction expansions that the convergents are the "best approximants", to a large class of continued fraction expansions of complex numbers.