Geometry of planar curves intersecting many lines in a few points

TL;DR

This paper investigates the geometric properties of planar curves that intersect many lines in few points, establishing strong Lipschitz regularity and measure estimates under specific directional constraints.

Contribution

It introduces a new Lipschitz regularity result for curves avoiding multiple intersections with lines in a cone, surpassing classical theorems in strength.

Findings

Lipschitz property for graphs avoiding multiple line intersections

Curve decomposition with finite Hausdorff measure

Explicit Hausdorff measure estimates for curve segments

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.