The dimension of projections of fractal percolations

TL;DR

This paper investigates the geometric measure properties of fractal percolations, showing that projections preserve dimension below 1 and images contain intervals when dimension exceeds 1, revealing regularity in these random fractals.

Contribution

It establishes the dimension of projections and the existence of intervals in images of fractal percolations, connecting geometric measure theory with the structure of these random sets.

Findings

Projections of sets with Hausdorff dimension less than 1 retain the same dimension.

Images of sets with Hausdorff dimension greater than 1 contain intervals under certain functions.

The results apply to algebraic sums of multiple fractal percolations.

Abstract

\emph{Fractal percolation} or \emph{Mandelbrot percolation} is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by $E\subset \mathbb{R}^2$ a typical realization of the fractal percolation on the plane, {itemize} If $\dim_{\rm H}E<1$ then for \textbf{all}lines $\ell$ the orthogonal projection $E_\ell$ of $E$ to $\ell$ has the same Hausdorff dimension as $E$, If $\dim_{\rm H}E>1$ then for any smooth real valued function $f$ which is strictly increasing in both coordinates, the image $f(E)$ contains an interval. {itemize} The second statement is quite interesting considering the fact that $E$ is almost surely a Cantor set (a {\it random dust}) for a large part of the parameter domain, see \cite{Chayes1988}. Finally, we solve a related problem about the existence of an interval in the algebraic sum of $d\geq 2$ one-dimensional fractal percolations.