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 $$R_{\mathit{a}}=I_{a_1}\times \cdots \times I_{a_{n-1}}\times I_{-(a_1+\cdots +a_{n-1})}$$ for any $\mathit{a}=(a_1,\dots ,a_{n-1})\in A^{n-1}$ 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 $L^1{(1+{\log }^{+}{L}^1)}^{n-1}$ to $L^{1,\mathrm{\infty }}$.