Nikod\'ym maximal function with restricted directions

We study the planar Nikod\'ym maximal operator $\mathcal{N}_{\Theta;\delta}$ associated to a direction set $\Theta \subset \mathbb{S}^{1}$. We show that the quasi-Assouad dimension $s := \dim_{\mathrm{qA}} \Theta$ characterises the essential $L^{p}$-boundedness of $\mathcal{N}_{\Theta;\delta}$ in the following sense. If $s \in [\tfrac{1}{2},1]$, then $\mathcal{N}_{\Theta;\delta}$ is essentially bounded on $L^{p}(\mathbb{R}^{2})$ for $p \geq 1 + s$, and essentially unbounded for $p < 1 + s$. Here essential boundedness means $L^{p}$-boundedness with constant $O_{\epsilon}(\delta^{-\epsilon})$. We also show that the characterisation described above fails for $s < \tfrac{1}{2}$. More precisely, there exists a set $\Theta \subset \mathbb{S}^{1}$ with $\dim_{\mathrm{qA}} \Theta = \tfrac{1}{3}$ such that $\mathcal{N}_{\Theta;\delta}$ is essentially unbounded on $L^{p}(\mathbb{R}^{2})$ for all $p < \tfrac{3}{2}$. As an application, we show there exists a convex domain with affine dimension $\tfrac{1}{6}$ such that the $\alpha$-order Bochner-Riesz means converge in $L^6$ for all $\alpha>0$.