Probabilistic Construction of Kakeya-Type Sets in $\mathbb{R}^2$ associated to separated sets of directions

TL;DR

This paper establishes conditions on direction sets in the plane that lead to unboundedness of the associated directional maximal operator on all Lebesgue spaces, using probabilistic methods inspired by Kakeya set constructions.

Contribution

It introduces a new probabilistic framework for constructing Kakeya-type sets related to separated directions, demonstrating unboundedness of the maximal operator.

Findings

Directional maximal operator is unbounded for certain direction sets

Probabilistic construction techniques extend previous Kakeya set ideas

Results apply to all p in [1, ∞)

Abstract

We provide a condition on a set of directions $\Omega \subset \mathbb{S}^1$ ensuring that the associated directional maximal operator $M_\Omega$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. The techniques of proof extend ideas of Bateman and Katz involving probabilistic construction of Kakeya-type sets involving sticky maps and Bernoulli percolation.