A sharp estimate for the Hilbert transform along finite order lacunary sets of directions

Let $D$ be a nonnegative integer and ${\mathbf{\Theta}}\subset S^1$ be a lacunary set of directions of order $D$. We show that the $L^p$ norms, $1<p<\infty$, of the maximal directional Hilbert transform in the plane $$H_{{\mathbf{\Theta}}} f(x):= \sup_{v\in {\mathbf{\Theta}}} \Big|\mathrm{p.v.}\int_{\mathbb R }f(x+tv)\frac{\mathrm{d} t}{t}\Big|, \qquad x \in {\mathbb R}^2,$$ are comparable to $(\log\#{\mathbf{\Theta}})^\frac{1}{2}$. For vector fields $\mathsf{v}_D$ with range in a lacunary set of of order $D$ and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field $\mathsf{v}_D$, $$H_{\mathsf{v}_D,1} f(x):= \mathrm{p.v.} \int_{ |t| \leq 1 } f(x+t\mathsf{v}_D(x)) \,\frac{\mathrm{d} t}{t},$$ is $L^p$-bounded for all $1<p<\infty$. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.