Universal singular sets and unrectifiability (Q1950973)

Summary: The geometry of universal singular sets has recently been studied by \textit{M. Csörnyei} et al. [``Universal singular sets in the calculus of variations'', Arch. Ration. Mech. Anal. 190, No. 3, 371--424 (2008)]. 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.