Hyperplane absolute winning property of bounded orbits under diagonalizable flows on $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$

Abstract

We extend the work of An, Guan and Kleinbock on bounded orbits of diagonalizable flows on $\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$ to $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$, where $\mathbb{K}$ is an imaginary quadratic field. To achieve this, we first prove a complex analogue of Minkowski's Linear Forms Theorem. We then set up an appropriate Schmidt game in $\mathbb{C}^3$ such that bounded orbits correspond to a hyperplane-absolute-winning set consisting of certain vectors in $\mathbb{C}^3$ relative to an approximation by imaginary quadratic rationals in $\mathbb{K}$.