On the hitting probabilities of limsup random fractals

TL;DR

This paper investigates the probability that a limsup random fractal intersects with an analytic set, extending previous results by relaxing homogeneity conditions and exploring the role of Hausdorff versus packing dimensions.

Contribution

It generalizes existing hitting probability results for limsup random fractals by removing homogeneity assumptions and analyzing the significance of Hausdorff dimension.

Findings

Hitting probability depends on Hausdorff dimension exceeding a threshold

Extension of prior work to non-homogeneous probability models

Counterexamples show Hausdorff dimension condition cannot be replaced by packing dimension

Abstract

Let $A$ be a limsup random fractal with indices $\gamma_1, ~\gamma_2 ~$and $\delta$ on $[0,1]^d$. We determine the hitting probability $\mathbb{P}(A\cap G)$ for any analytic set $G$ with the condition $(\star)$$\colon$ $\dim_{\rm H}(G)>\gamma_2+\delta$, where $\dim_{\rm H}$ denotes the Hausdorff dimension. This extends the correspondence of Khoshnevisan, Peres and Xiao [10] by relaxing the condition that the probability $P_n$ of choosing each dyadic hyper-cube is homogeneous and $\lim\limits_{n\to\infty}\frac{\log_2P_n}{n}$ exists. We also present some counterexamples to show the Hausdorff dimension in condition $(\star)$ can not be replaced by the packing dimension.