# Universal Singular Sets and Unrectifiability

### Richard Gratwick

St. John's College, Oxford, UK

## Abstract

The geometry of universal singular sets has recently been studied by M.~~Cs\"ornyei 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.

## Cite this article

Richard Gratwick, Universal Singular Sets and Unrectifiability. Z. Anal. Anwend. 32 (2013), no. 2, pp. 179–197

DOI 10.4171/ZAA/1480