  Let $\Omega $ be any set of directions (unit vectors) on the plane.
  In this paper we study maximal operator of the one dimensional maximal
function computed in the directions of $\Omega$
  We are interested in extensions of lacunary sets of directions, to
collections we call $N$--lacunary, for integers $N$. We proceed by induction.
Say that $\Omega$ is 1--lacunary iff $\Omega$ is an ordinary lacunary set of
vectors.
  Every $N+1$--lacunary set can be obtained from some $N$--lacunary $\Omega_N$
adding some points to $\Omega_N$. Between each two neighbor points
$a,b\in\Omega_N$ we can add a 1--lacunary sequence (finite or infinite).
  We show that for all $N$ lacunary sets $\Omega$, $$ \|M_\Omega
f(x)\|_2\lesssim{}N \|f\|_2. $$ Observe that every set $\Omega$ of $N$ points
is $(C\log N)$--lacunary. We then obtain a Theorem of N. Katz \cite{Katz2}.
Both the current inequality, and Katz' result are consequence of a general
result of Alfonseca, Soria, and Vargas \cites{ASV2}. We offer the current proof
as a succinct, self--contained approach to this inequality.
