低密度Cantor集与$C^1$曲线上的Lipschitz函数
Cantor sets of low density and Lipschitz functions on $C^1$ curves
摘要：
本文刻划了函数$f\colon [0,1] \longrightarrow [0,1]$存在正测度的可测集$C\subseteq [0,1]$满足对任意非平凡区间$I \subseteq [0,1]$的$\frac{|C\cap I|}{|I|}<f(|I|)$的性质.作为一个应用，我们证明了在任何$C^1$曲线，它是可能的，以构建一个Lipschitz函数，不能近似Lipschitz函数达到其Lipschitz常数。
英文摘要：
We characterize the functions $f\colon [0,1] \longrightarrow [0,1]$ for which there exists a measurable set $C\subseteq [0,1]$ of positive measure satisfying $\frac{|C\cap I|}{|I|}<f(|I|)$ for any nontrivial interval $I \subseteq [0,1]$. As an application, we prove that on any $C^1$ curve it is possible to construct a Lipschitz function that cannot be approximated by Lipschitz functions attaining their Lipschitz constant.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
