Gauss lattices and complex continued fractions

TL;DR

This paper introduces a complex continued fraction algorithm based on Gaussian integer lattices that efficiently finds all best Diophantine approximations to a complex number, extending classical approximation theory.

Contribution

It develops a novel algorithm using minimal vectors in Gaussian integer lattices, linking lattice theory with complex Diophantine approximation.

Findings

Algorithm finds all best approximations to a complex number.

Determines the best constant for complex Dirichlet Theorem.

Establishes a correspondence between minimal vectors and best approximations.

Abstract

Our aim is to find a complex continued fraction algorithm finding all the best Diophantine approximations to a complex number. Using the sequence of minimal vectors in a two dimensional lattice over Gaussian integers, we obtain an algorithm defined on a submanifold of the space of unimodular two dimensional Gauss lattices. This submanifold is transverse to the diagonal flow. Thanks to the correspondence between minimal vectors and best Diophantine approximations, the algorithm finds all the best approximations to a complex number. A byproduct of the algorithm is the best constant for the complex version of Dirichlet Theorem about approximations of complex numbers by quotients of Gaussian integers.