Differentiation properties of class $L^1([0,1{]}^2)$ with respect to two different basis of rectangles

It is a well-known result by Saks \cite{Saks1934} that there exists a function $f\in L^1({\mathbb{R}}^2)$ so that for almost every $(x,y)\in $\mathbb{R}$^2$ $$\underset{\begin{array}{c}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}R\to 0,\\ (x,y)\in R\in \mathcal{R}\end{array}}{\lim }\mid \frac{1}{\mathrm{\mid }R\mathrm{\mid }}{\int }_{R}f(x,y)dxdy\mid =\mathrm{\infty },$$ where $\mathcal{R}=\{[a,b)\times [c,d)\text{\hspace{0.17em}⁣}:a<b,c<d\}$. In this note we address the following question: assume we have two different collections of rectangles; under which conditions there exists a function $f \in L^1($\mathbb{R}$^2)$ so that its integral averages are divergence with respect to one collection and convergence with respect to another? More specifically, let $\mathcal{D},\mathcal{C}\subset (0,1]$ and consider rectangles with side lengths in $\mathcal{D}$ and respectively in $\mathcal{C}$. We show that if the sets $\mathcal{D}$ and $\mathcal{C}$ are sufficient ``far" from each other, then such a function can be constructed. We also show that in the class of positive functions our condition is also necessary for such a function to exist.