Cantor sets of low density and Lipschitz functions on $C^1$ curves

Abstract

We characterize the functions $f:[0,1]⟶[0,1]$ 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.

11 pages, 1 figure