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

Gratwick R. Universal Singular Sets and Unrectifiability. Z. Anal. Anwend. 32 (2013), 179-197. doi: 10.4171/ZAA/1480