Best Diophantine approximations in the complex plane with Gaussian integers

Abstract: Starting with the minimal vectors in lattices over Gaussian integers in $\C^2$, we define a algorithm that finds the sequence of minimal vectors of any unimodular lattice in $\C^2$. Restricted to lattices associated with complex numbers this algorithm find all the best Diophantine approximations of a complex numbers. Following Doeblin, Lenstra, Bosma, Jager and Wiedijk, we study the limit distribution of the sequence of products $(u_{n1}u_{n2})_n$ where $(u _n=( u_{n1},u_{n2} ))_n$ is the sequence of minimal vectors of a lattice in $C^2$. We show that there exists a measure in $\C$ which is the limit distribution of the sequence of products of almost all unimodular lattices.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper )

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