Diophantine approximations on random fractals

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 realization of a fractal percolation process, then almost surely (conditioned on $E\ne \varnothing$), 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.

Some very minor corrections, added some figures

References in corpus (1)

• Badly approximable vectors and fractals defined by conformal dynamical systems