Cantor sets of low density and Lipschitz functions on C1 curves

Abstract

We characterize the functions f:(0,1]⟶[0,1] for which there exists a measurable set C⊆[0,1] of positive measure satisfying |C∩I||I|<f(|I|) for any nontrivial interval I⊆[0,1]. As an application, we prove that on any injective C1 curve it is possible to construct a Lipschitz function that cannot be approximated by Lipschitz functions attaining their Lipschitz constant. Finally, we extend this result to more general C1 curves.