We provide a condition on a set of directions Ω ⊂ S¹ ensuring that the associated directional maximal operator M_Ω is unbounded on Lᵖ(R²) for every 1 ≤ p < ∞. 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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Hagelstein et al. (2024) studied this question.
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