Zeitschrift für Analysis und ihre Anwendungen

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Volume 32, Issue 2, 2013, pp. 179–197
DOI: 10.4171/ZAA/1480

Universal Singular Sets and Unrectifiability

Richard Gratwick (1)

(1) Department of Mathematics, St. John's College, OX1 3JP, OXFORD, UNITED KINGDOM

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.

Keywords: Partial regularity, universal singular set, purely unrectifiable set