$\mathbf{Bad}(\mathbf{r};\mathbf{s})$ is Hyperplane Absolute Winning

Abstract

Given an $m$-dimensional weight $\mathbf{r}$ and an $n$-dimensional weight $\mathbf{s}$, we prove that the set of $(\mathbf{r};\mathbf{s})$-badly approximable $m\times n$ matrices is hyperplane absolute winning on ${\mathbb{R}}^{m\times n}$. This fully answers a question \cite[Question 8.2 (iii)]{Kl} of D. Kleinbock in 1998.

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