Pure and Applied Mathematics Quarterly
Volume 17 (2021)
Gauss lattices and complex continued fractions
Pages: 1785 – 1860
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.
Gaussian integer, lattice, best Diophantine approximation
2010 Mathematics Subject Classification
Primary 11J04, 11J70. Secondary 11Hxx.
Received 13 January 2021
Received revised 27 October 2021
Accepted 4 November 2021
Published 26 January 2022