Sharp weak-type estimate for maximal operators associated to Cartesian families under an arithmetic condition

Given a set of integers $A \subset \mathbb{Z}$, we consider the smallest family $\mathcal{B}_{A^{n-1}}$ invariant by translation which contains the rectangles $$R_{\boldsymbol{a}} = I_{a_1} \times \dots \times I_{a_{n-1}} \times I_{-(a_1+\dots+a_{n-1})}$$ for any $\boldsymbol{a} = (a_1,\dots,a_{n-1}) \in A^{n-1}$ and where $I_k = [0,2^k]$ for $k$ integer. We prove that if the set $A$ contains arbitrary large arithmetic progression then the maximal operator $M_{\mathcal{B}_{A^{n-1}}}$ associated to the family $\mathcal{B}_{A^{n-1}}$ is sharply bounded from $L^1\left(1+ \log^+ L^1 \right)^{n-1}$ to $L^{1,\infty}$.