Continued fractions for complex numbers and values of binary quadratic forms

TL;DR

This paper explores properties of continued fraction expansions of complex numbers using Gaussian integers, introduces new algorithms and sequences, and applies findings to binary quadratic forms with dense value sets.

Contribution

It introduces a generalized framework for continued fractions of complex numbers, including new algorithms and characterizations, extending classical results to the complex domain.

Findings

Multiple continued fraction expansions for complex numbers are possible.

Analogues of Lagrange's theorem are established for complex quadratic surds.

Binary quadratic forms with complex coefficients can have dense value sets over Gaussian integers.

Abstract

We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as also in a more general way from what we call "iteration sequences". We consider in this broader context the analogues of the Lagrange theorem characterizing quadratic surds, the growth properties of the denominators of the convergents, and the overall relation between sequences satisfying certain conditions, in terms of nonoccurrence of certain finite blocks, and the sequences involved in continued fraction expansions. The results are also applied to describe a class of binary quadratic forms with complex coefficients whose values over the set of pairs of Gaussian integers form a dense set of complex numbers.