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

TL;DR

This paper characterizes functions related to measure-theoretic properties of sets and demonstrates the construction of special Lipschitz functions on $C^1$ curves that defy approximation by Lipschitz functions with the same Lipschitz constant.

Contribution

It provides a characterization of functions linked to measure conditions and constructs Lipschitz functions on $C^1$ curves with unique approximation properties.

Findings

Characterization of functions with measure-based properties.

Existence of Lipschitz functions on $C^1$ curves that cannot be approximated by similar Lipschitz functions.

Application to measure and Lipschitz function theory.

Abstract

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.