Projections of fractal percolations

TL;DR

This paper investigates the geometric properties of Mandelbrot percolation fractals, showing that under certain conditions, their projections and distance sets almost surely contain intervals, revealing rich structure in these random fractals.

Contribution

It proves that for Mandelbrot percolation sets with Hausdorff dimension greater than 1, all projections and distance sets almost surely contain intervals, extending understanding of their geometric complexity.

Findings

Orthogonal projections to every line contain intervals.

Radial projections from every center contain intervals.

Distance sets from every point contain intervals.

Abstract

In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset \mathbb{R}^2 $ which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than 1 then the orthogonal projection to \textbf{every} line, the radial projection with \textbf{every} center, and distance set from \textbf{every} point contain intervals.