AI 中文总结
本文针对ℝⁿ中的非退化流形开发新纤维化机制，证明其可分解为具一致定量非退化性质的曲线族，该结构在小Cⁿ微扰下稳定，为丢番图逼近结果从曲线扩展到流形提供框架。
AI 中文摘要
受丢番图逼近应用的启发，我们为ℝⁿ中的非退化流形开发了一种新的纤维化机制，证明它们可局部分解为具有一致定量非退化性质的非退化曲线族。这些流形需为Cⁿ类，且满足本质最优界N≥max{n,l+1}，其中该流形为l非退化的。此外，我们证明这种一致定量的局部控制可全局化为一个定性结论，即非退化流形的几乎每个点都位于该流形包含的一条非退化曲线上。我们的方法是微扰性的，表明解析情形下产生的纤维化结构在足够小的Cⁿ微扰下是稳定的。该方法提供了一个框架，可将丢番图逼近中基于纤维化的结果从曲线扩展到流形，本文中考虑了此类应用。
英文摘要
Motivated by applications in Diophantine approximation, we develop a new fibering mechanism for non-degenerate manifolds in $\mathbb{R}^n$, showing that they admit local decompositions into families of non-degenerate curves with uniform quantitative non-degeneracy properties. The manifolds are required to be $C^N$ with the essentially optimal bound $N\geq \max\lbrace n,l+1\rbrace$, where the manifold is $l$ non-degenerate. Furthermore, we show that this uniform quantitative local control can be globalised into a qualitative statement, so that almost every point of a non-degenerate manifold lies on a non-degenerate curve contained in the manifold. Our approach is perturbative and shows that the fibering structure arising in the analytic setting is stable under sufficiently small $C^N$ perturbations. The method provides a framework for extending fibering based results in Diophantine approximation from curves to manifolds. Such applications are considered in the paper.
Comments59 pages
