Visible parts of fractal percolation

TL;DR

This paper investigates the dimensional properties of visible parts of fractal percolation in the plane, showing that under certain conditions, these parts are almost surely 1-dimensional with positive finite Hausdorff measure.

Contribution

It provides new results on the dimension and measure of visible parts of fractal percolation, addressing an open problem in the field.

Findings

Visible parts from lines are almost surely 1-dimensional when the fractal dimension is at least 1.

These visible parts have positive and finite Hausdorff measure.

Results also hold for visible parts from points.

Abstract

We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from lines are 1-dimensional. Furthermore, almost all of them have positive and finite Hausdorff measure. We also verify analogous results for visible parts from points. These results are motivated by an open problem on the dimensions of visible parts.