  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.
