Fractal Percolation,Porosity, and Dimension

We study the porosity properties of fractal percolation sets \(E\subset \mathbb {R}^d\). Among other things, for all \(0<\varepsilon <\tfrac{1}{2}\), we obtain dimension bounds for the set of exceptional points where the upper porosity of E is less than \(\tfrac{1}{2}-\varepsilon \), 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.