Projections of the random Menger sponge

TL;DR

This paper investigates the geometric properties of the random Menger sponge, a fractal generated by a stochastic process similar to fractal percolation, focusing on its projections and measure-theoretic features.

Contribution

It extends the analysis of random self-similar fractals by deriving results on projections, measure, and interior points for the random Menger sponge, using a novel probabilistic approach.

Findings

Projections have Hausdorff dimension consistent with the deterministic case.

Certain projections possess positive Lebesgue measure.

The random Menger sponge can have interior points under specific conditions.

Abstract

Using a similar random process to the one which yields the fractal percolation sets, starting from the deterministic Menger sponge we get the random Menger sponge. We examine its orthogonal projections from the point of Hausdorff dimension, Lebesgue measure and existence of interior points. We obtain these results as special cases of our theorems stated for random self-similar IFSs. These are obatained by a random process similar to the fractal percolation, applied for the cylinder sets of a deterministic self-similar IFS, as in arXiv:1212.1345. In this paper the associated deterministic IFS on the line is of the special form $\mathcal{S}=\left\{\frac{1}{L}x+t_i \right\} _{i=1}^{m}$, where $L\in\mathbb{N}$, $L\geq 2$ and $t_i\in\mathbb{Q}$.