Asian Journal of Mathematics

Volume 27 (2023)

Number 6

Projections of the random Menger sponge

Pages: 893 – 936

DOI: https://dx.doi.org/10.4310/AJM.2023.v27.n6.a4

Authors

Vilma Orgoványi (HUN-REN–BME Stochastics Research Group, Budapest, Hungary; Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Budapest, Hungary)

Károly Simon (HUN-REN–BME Stochastics Research Group, Budapest, Hungary; Alfréd Rényi Institute of Mathematics – Eötvös Loránd Research Network, Budapest, Hungary)

Abstract

In this paper we prove theorems about a special family of random self-similar sets on the line, we apply these theorems to get the Hausdorff dimension, the Lebesgue measure and existence of interior points of some projections of the random right-angled Sierpiski gasket, the random Sierpiński carpet and the random Menger sponge. The Menger sponge is one of the most well-known example of self-similar sets in $\mathbb{R}^3$. The Mandelbrot percolation process restricted to the cubes, which are the building blocks of the Menger sponge, yields the random Menger sponge, a random self-similar fractal in $\mathbb{R}^3$. We examine its orthogonal projections to straight lines, from the point of Lebesgue measure and existence of interior points. In particular this yields random self-similar sets on the line with positive Lebesgue measure and empty interior. Moreover, we give a sharp threshold for the probability above which the projections of the random Menger sponge contains an interval in all directions.

Keywords

random fractals, Mandelbrot percolation, branching processes in random environments

2010 Mathematics Subject Classification

Primary 28A80. Secondary 60J85.

Received 31 December 2022

Accepted 18 September 2023

Published 7 August 2024