Gauss lattices and complex continued fractions

Our aim is to construct 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 the ring of 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. The correspondence between the minimal vectors and the best Diophantine approximations ensures that our algorithm reaches its goal. A byproduct of the algorithm is the best constant for the complex version of Dirichlet’s theorem about approximations of complex numbers by quotients of two Gaussian integers.

Keywords

Gaussian integer, lattice, best Diophantine approximation

2010 Mathematics Subject Classification

Primary 11J04, 11J70. Secondary 11Hxx.

Full Text (PDF format)

Received 13 January 2021

Received revised 27 October 2021

Accepted 4 November 2021

Published 26 January 2022