Diophantine approximations on random fractals

TL;DR

This paper proves that certain random fractals in Euclidean space almost surely intersect all hyperplane absolute winning sets with full Hausdorff dimension, revealing deep connections between fractal geometry, Diophantine approximation, and probabilistic methods.

Contribution

It introduces a novel approach using Galton-Watson trees to analyze intersections of random fractals with Diophantine sets, extending results to a broad class of random fractals beyond traditional models.

Findings

Fractal percolation sets almost surely intersect every HAW set with full Hausdorff dimension.

Existence of hyperplane diffuse, Ahlfors-regular subsets within fractal percolation sets.

Method applicable to general random fractals generated by similarity IFS, not contained in a single hyperplane.

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 $E\neq\emptyset$), 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.