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Perron’s capacity of random sets
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 4 / November 2023
- Published online by Cambridge University Press:
- 01 September 2023, pp. 960-970
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- Article
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We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, 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 directionsand
\begin{equation*}\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}\end{equation*}
\begin{equation*}\Omega_{\text{rand},\text{lac}} := \left\{\frac{\pi X_k}{2^k} : k\geq 1 \right\}.\end{equation*}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 
$1 \lt p \lt \infty$.
