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Geometry of planar curves intersecting many lines in a few points

D. Vardakis^a, A. Volberg<sup>ab

a</sup> Department of Mathematics, Michigan State University, East Lansing, MI. 48823
^b Hausdorff Center for Mathematics, Bonn, Germany

Abstract: The local Lipschitz property is shown for the graph avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand's well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is $\sigma$-finite with respect to Hausdorff length and an estimate on the Hausdorff measure of each “piece” is found.

Keywords: Hausdorff dimension, Lipschitz function, Marstrand's theorem.

Received: 15.03.2021

Language: English

 English version: St. Petersburg Mathematical Journal, 2022, 33:6, 1047–1062