Weighted badly approximable complex vectors and bounded orbits of certain diagonalizable flows

TL;DR

This paper extends results on badly approximable vectors to complex number fields, showing that certain bounded orbit sets are hyperplane-absolute-winning, and generalizes previous results on badly approximable complex numbers.

Contribution

It introduces a new analogue of a theorem for totally imaginary number fields and demonstrates that bounded orbit sets form hyperplane-absolute-winning sets in this context.

Findings

Bounded orbits form hyperplane-absolute-winning sets.

Generalization of badly approximable complex numbers.

Extension of results to totally imaginary number fields.

Abstract

We show an analogue of a theorem of An, Ghosh, Guan, and Ly on weighted badly approximable vectors for totally imaginary number fields. We show that for $G=\mathrm{SL}_2(\mathbb{C})\times\dots\times\mathrm{SL}_2(\mathbb{C})$ and $\Gamma<G$ a lattice subgroup, the points of $G/\Gamma$ with bounded orbits under a one-parameter Ad-semisimple subgroup of $G$ form a hyperplane-absolute-winning set. As an application, we also provide a generalization of a result of Esdahl-Schou and Kristensen about the set of badly approximable complex numbers.