Minimal vectors in $\C^2$ and best constant for Dirichlet theorem over $\C$

Abstract: We study minimal vectors in lattices over Gaussian integers in $\C^2$.We show that the index of the sub-lattice generated by two consecutive minimal vectors in a lattice of $\C^2$, can be either $1$ or $2$.Next, we describe the constraints on pairs of consecutive minimal vectors. These constraints make it possible to find the best constant for Dirichlet theorem about approximations of complex numbers by quotient of Gaussian integers.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar