Abstract
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 realisation 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 analysing 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.
Hausdorff dimension of limsup random fractals
