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

Chengyang Wu

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arXiv · 2609.22016

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

Chengyang Wu

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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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Chengyang Wu. 2026-09-18. $\mathbf{Bad}(\mathbf{r};\mathbf{s})$ is Hyperplane Absolute Winning. https://arxiv.org/abs/2609.22016

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