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It is a well-known result by Saks \cite{Saks1934} that there exists afunction $f \in L^1(\mathbb{R}^2)$ so that for almost every $(x,y)\in\mathbb{R}^2$ \[ \lim_{\substack{\mathrm{diam} R\rightarrow 0, \\ (x,y) \in R\in \mathcal{R}}}\left|\frac{1}{|R|}\int_R f(x,y)\, dxdy\right|=\infty, \]where $\mathcal{R}=\{[a,b)\times [c,d)\colon a<b, c<d\}$. In this note weaddress the following question: assume we have two different collections ofrectangles; under which conditions there exists a function $f \inL^1(\mathbb{R}^2)$ so that its integral averages are divergence with respect toone collection and convergence with respect to another? More specifically, let$\mathcal{D}, \mathcal{C} \subset (0,1]$ and consider rectangles with sidelengths in $\mathcal{D}$ and respectively in $\mathcal{C}$. We show that if thesets $\mathcal{D}$ and $\mathcal{C}$ are sufficient ``far" from each other,then such a function can be constructed. We also show that in the class ofpositive functions our condition is also necessary for such a function toexist.

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