Universal Singular Sets and Unrectifiability

The geometry of universal singular sets has recently been studied by M. Csornyei et al. [Arch. Ration. Mech. Anal. 190 (2008)(3), 371-424]. In particular they proved that given a purely unrectifiable compact set S subset of R-2, one can construct a C-infinity-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-infinity-Lagrangian, of arbitrary superlinearity, with universal singular set covering this subset.