Remarks on the construction of $K_σ$ sets associated to trees not satisfying a separation condition

Abstract

$K_σ$ sets involving sticky maps $σ$ 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 $1 \leq p < \infty$. We indicate limits to this approach by showing that, given $ε> 0$ and a natural number $N$, there exists a tree $\mathcal{T}_{N, ε}$ of finite height that is lacunary of order $N$ but such that, for \emph{every} sticky map $σ: \mathcal{B}^{h(\mathcal{T}_{N, ε})} \rightarrow \mathcal{T}_{N, ε}$, one has $|K_σ \cap ((1,2) \times \mathbb{R})| \geq 1 - ε$.