Fractal percolation, porosity, and dimension

We study the porosity properties of fractal percolation sets $E\subset {\mathbb{R}}^d$. Among other things, for all $0<\epsilon <\frac{1}{2}$, we obtain dimension bounds for the set of exceptional points where the upper porosity of $E$ is less than $\frac{1}{2}-\epsilon$, or the lower porosity is larger than $\epsilon$. Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton-Watson process.