Application of Perron trees to geometric maximal operators

We characterize the $L^p(\mathbb R^2)$ boundedness of the geometric maximal operator $M_{a,b}$ associated to the basis $\mathcal B_{a,b}$ ($a,b \gt 0$) which is composed of rectangles $R$ whose eccentricity and orientation are 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 generalized Perron trees, as constructed by Hare and Röning [J. Fourier Anal. Appl. 4 (1998)].