  We characterize the $L^p(\mathbb{R}^2)$ boundeness of the geometric maximal
operator $M_{a,b}$ associated to the basis $\mathcal{B}_{a,b}$ ($a,b > 0$)
which is composed of rectangles $R$ whose eccentricity and orientation is of
the form $$\left( e_R ,\omega_R \right) = \left( \frac{1}{n^a} ,
\frac{\pi}{4n^b} \right)$$ for some $n \in \mathbb{N}^*$. The proof involves
\textit{generalized Perron trees}, as constructed in \cite{KATHRYN JAN}.
