Universal Singular Sets and Unrectifiability

Abstract

The geometry of universal singular sets has recently been studied by M. Csörnyei et al. [Arch. Ration. Mech. Anal. 190 (2008)(3), 371–424]. In particular they proved that given a purely unrectifiable compact set $S\subseteq {\mathbb{R}}^2$, one can construct a $C^{\infty }$-Lagrangian with a given superlinearity such that the universal singular set of $L$ contains $S$. We show the natural generalization: That given an $F_{\sigma }$ purely unrectifiable subset of the plane, one can construct a $C^{\infty }$-Lagrangian, of arbitrary superlinearity, with universal singular set covering this subset.