Continued fraction expansions of complex numbers, Lagrange's theorem, and badly approximable numbers

TL;DR

This paper extends Lagrange's theorem to complex numbers using a generalized continued fraction approach, linking it to quadratic and Hermitian forms, and explores implications for badly approximable numbers.

Contribution

It introduces a broader framework for continued fractions in complex numbers, generalizes Lagrange's theorem, and connects it to matrix sequences and badly approximable numbers.

Findings

Generalized Lagrange theorem for complex numbers

Matrix-based formulation of continued fractions

Extended results on badly approximable complex numbers

Abstract

This paper concerns extension of the classical Lagrange theorem, on the eventual periodicity of continued fraction expansions of quadratic surds, and the versions of it found in the literature in the case of complex numbers. In this respect, firstly, we adopt a more general notion of continued fraction expansions, in place of those arising from the nearest integer algorithms. Secondly, the issue is formulated in terms of zeros of quadratic and Hermitian forms, and a result is proved in terms of certain sequences of matrices associated with them, via continued fraction expansions. The result may be considered as a matrix analogue of Lagrange's theorem in the general framework. The unified approach leads to generalizations of the Lagrange theorem on one hand, and an extended version of a result of Hines (2019) on badly approximable complex numbers, on the other hand.