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

Abstract

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

Includes minor corrections to original submission. To appear in Duke Math. J