  We establish the sharp growth rate, in terms of cardinality, of the $L^p$
norms of the maximal Hilbert transform $H_\Omega$ along finite subsets of a
finite order lacunary set of directions $\Omega \subset \mathbb R^3$, answering
a question of Parcet and Rogers in dimension $n=3$. Our result is the first
sharp estimate for maximal directional singular integrals in dimensions greater
than 2.
  The proof relies on a representation of the maximal directional Hilbert
transform in terms of a model maximal operator associated to compositions of
two-dimensional angular multipliers, as well as on the usage of weighted norm
inequalities, and their extrapolation, in the directional setting.
