The geometry of universal singular sets has recently been studied by M. Cs?rnyei et al. [Arch. Ration. Mech. Anal. 190 (2008)(3), 371-424]. In particular they proved that given a purely unrectifiable compact set S ? R~2, one can construct a C~∞-Lagrangian with a given superlinearity such that the universal singular set of L contains S. We show the natural generalization: That given an F_σ purely unrectifiable subset of the plane, one can construct a C~∞-Lagrangian, of arbitrary superlinearity, with universal singular set covering this subset.
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