Fractal percolation, porosity, and dimension

TL;DR

This paper investigates the porosity and dimension properties of fractal percolation sets, providing bounds for exceptional points and extending results to inhomogeneous and more general random fractal sets.

Contribution

It introduces new dimension bounds for exceptional points in fractal percolation and extends the analysis to inhomogeneous and general Galton-Watson type random sets.

Findings

Dimension bounds for points with specific porosity properties

Extension of results to inhomogeneous fractal percolation

Applicability to general Galton-Watson process-based sets

Abstract

We study the porosity properties of fractal percolation sets $E\subset\mathbb{R}^d$. Among other things, for all $0<\varepsilon<\tfrac12$, we obtain dimension bounds for the set of exceptional points where the upper porosity of $E$ is less than $\tfrac12-\varepsilon$, or the lower porosity is larger than $\varepsilon$. Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton-Watson process.