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In this paper, we study Lebesgue differentiation processes along rectangles$R_k$ shrinking to the origin in the Euclidean plane, and the question of theiralmost everywhere convergence in $L^p$ spaces. In particular, classes ofexamples of such processes failing to converge a.e. in $L^\infty$ are provided,for which $R_k$ is known to be oriented along the slope $k^{-s}$ for $s>0$,yielding an interesting counterpart to the fact that the directional maximaloperator associated to the set $\{k^{-s}:k\in\mathbb{N}^*\}$ fails to bebounded in $L^p$ for any $1\leq p<\infty$.

Almost everywhere convergence for Lebesgue differentiation processes along rectangles