Diophantine approximations on random fractals

We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realization of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$. We show this by proving that $E$ almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$. We achieve this by analyzing Galton-Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of $E$. This method allows us to obtain a more general result by projecting the Galton-Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.