Hausdorff dimension of Besicovitch sets of Cantor graphs

TL;DR

This paper investigates the Hausdorff dimension of planar Besicovitch sets containing rotated copies of Cantor graphs, establishing a lower bound based on the Cantor set's dimension for a broad class of such sets.

Contribution

It introduces a new lower bound for the Hausdorff dimension of Besicovitch sets associated with Cantor graphs, expanding understanding of geometric measure properties of these fractal sets.

Findings

Lower bound of Hausdorff dimension: min(2 - s^2, 1/s)

Applicable to a large class of Cantor sets and graphs

Advances knowledge of fractal geometry in Besicovitch sets

Abstract

We consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets $\Gamma$, i.e. sets that contain a rotated copy of $\Gamma$ in each direction. We show that for a large class of Cantor sets $C$ and Cantor-graphs $\Gamma$ built on $C$, the Hausdorff dimension of any $\Gamma$-Besicovitch set must be at least $\min\left(2-s^2,\frac{1}{s}\right)$, where $s=\dim C$.