  We introduce the notion of \textit{Perron capacity} of a set of slopes
$\Omega \subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of
$\Omega$ is finite then the directional maximal operator associated $M_\Omega$
is not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$. This allows us
to prove that the set $$\Omega_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}:
n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question
raised by A. Stokolos.
