Perron's capacity of random sets

Abstract

Given a sequence of random variables $\left\{X_k:k\ge 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions $${\Omega }_{\text{rand},\text{lin}}:=\left\{\frac{\pi X_k}{k}:k\ge 1\right\}$$ and $${\Omega }_{\text{rand},\text{lac}}:=\left\{\frac{\pi X_k}{2^k}:k\ge 1\right\}.$$ 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 $1<p<\infty$.