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

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