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, $1<p<\mathrm{\infty }$, of the maximal directional Hilbert transform in the plane $$H_{\mathbf{\Theta }}f(x):=\underset{v\in \mathbf{\Theta }}{\sup }\mid \mathrm{p}\mathrm{.}\mathrm{v}\mathrm{.}{\int }_{\mathbb{R}}f(x+tv)\frac{\mathrm{d}t}{t}\mid ,x\in {\mathbb{R}}^2,$$ are comparable to $(\log \mathrm{\#}\mathbf{\Theta }{)}^{\frac{1}{2}}$. 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 ${\mathsf{v}}_D$, $$H_{{\mathsf{v}}_D,1}f(x):=\mathrm{p}\mathrm{.}\mathrm{v}\mathrm{.}{\int }_{\mathrm{\mid }t\mathrm{\mid }\le 1}f(x+t{\mathsf{v}}_D(x))\frac{\mathrm{d}t}{t},$$ is $L^p$-bounded for all $1<p<\mathrm{\infty }$. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.