  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 $L^{1,\infty}$.

Classical Analysis and ODEs