Key points are not available for this paper at this time.
We provide a condition on a set of directions S¹ ensuring that the associated directional maximal operator M_ 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.
Hagelstein et al. (Mon,) studied this question.