Perron's capacity of random sets
Anthony Gauvan · arXiv (Cornell University) · 2023
Given a sequence of random variables $\left\{ X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions $$Ω_{\text{rand},\text{lin}} := \left\{ \frac{πX_k}{k}: k \geq 1\right\}$$ and $$Ω_{\text{rand},\text{lac}} := \left\{ \frac{ πX_k}{2^k} : k\geq 1 \right\}.$$ We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$.