In this paper, we study the weighted n-dimensional badly approximable points on manifolds. Given a \(C^n\) differentiable non-degenerate submanifold \({\mathcal {U}} \subset {\mathbb {R}}^n\), we will show that any countable intersection of the sets of the weighted badly approximable points on \({\mathcal {U}}\) has full Hausdorff dimension. This strengthens a result of Beresnevich (Invent Math 202(3):1199–1240, 2015) by removing the condition on weights and weakening the smoothness condition on manifolds. Compared to the work of Beresnevich, our approach relies on homogeneous dynamics. It turns out that in order to solve this problem, it is crucial to study the distribution of long pieces of unipotent orbits in homogeneous spaces. The proof relies on the linearization technique and representations of \(\mathrm {SL}(n+1,{\mathbb {R}})\).

中文翻译：

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齐整空间中流形和单能轨道上的极近似点

在本文中，我们研究了流形上的加权n维重逼近点。给定一个\（C ^ n \）可微的非退化子流形\（{\ mathcal {U}} \ subset {\ mathbb {R}} ^ n \），我们将证明加权集合的任何可数交集\（{\ mathcal {U}} \）上的极近似点具有完整的Hausdorff尺寸。通过删除权重条件并削弱流形上的平滑度条件，可以增强Beresnevich的结果（Invent Math 202（3）：1199–1240，2015）。与Beresnevich的工作相比，我们的方法依赖于均匀动力学。事实证明，为了解决这个问题，研究均质空间中长单势轨道的分布至关重要。该证明依赖于线性化技术和\（\ mathrm {SL}（n + 1，{\ mathbb {R}}）\）的表示形式。