Remarks on the construction of $K_{\sigma }$ sets associated to trees not satisfying a separation condition

Abstract

$K_{\sigma }$ sets involving sticky maps $\sigma$ have been used in the theory of differentiation of integrals to probabilistically construct Kakeya-type sets that imply certain types of directional maximal operators are unbounded on $L^p({\mathbb{R}}^2)$ for all $1\le p<\infty$. We indicate limits to this approach by showing that, given $\epsilon >0$ and a natural number $N$, there exists a tree ${\mathcal{T}}_{N,\epsilon }$ of finite height that is lacunary of order $N$ but such that, for every sticky map $\sigma :{\mathcal{B}}^{h({\mathcal{T}}_{N,\epsilon })}\to {\mathcal{T}}_{N,\epsilon }$, one has $|K_{\sigma }\cap ((1,2)\times \mathbb{R})|\ge 1-\epsilon$.