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arXiv · 2405.17674

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

Paul Hagelstein·Blanca Radillo-Murguia·Alexander Stokolos
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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.

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Paul Hagelstein, Blanca Radillo-Murguia, Alexander Stokolos. 2024-05-27. Probabilistic Construction of Kakeya-Type Sets in $\mathbb{R}^2$ associated to separated sets of directions. https://arxiv.org/abs/2405.17674

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