(Un)boundedness of directional maximal operators through a notion of "Perron capacity'' and an application

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