Fractal Percolations

TL;DR

This paper explores the properties of fractal percolation sets, focusing on recent advances in the homogeneous case and discussing challenges and new findings in the non-homogeneous case.

Contribution

It reviews recent results on projection and slicing properties in homogeneous fractal percolation and presents new partial results for the non-homogeneous case.

Findings

Recent results on projection and slicing in homogeneous case

Discussion of difficulties in non-homogeneous case

New partial results in non-homogeneous fractal percolation

Abstract

One of the most well known random fractals is the so-called Fractal percolation set. This is defined as follows: we divide the unique cube in $\mathbb{R}^d$ into $M^d$ congruent sub-cubes. For each of these cubes a certain retention probability is assigned with which we retain the cube. The interior of the discarded cubes contain no points from the Fractal percolation set. In the retained ones we repeat the process ad infinitum. The set that remains after infinitely many steps is the Fractal percolation set. The homogeneous case is when all of these probabilities are the same. Recently, there have been considerable developments in regards with the projection and slicing properties in the homogeneous case. In the first part of this note we give an account of some of these recent results and then we discuss the difficulties and provide some new partial results in the non-homogeneous case.