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Diophantine approximations on random fractals. (arXiv:1807.05023v1 [math.PR])
来源于:arXiv
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 m 查看全文>>