Almost everywhere convergence for Lebesgue differentiation processes along rectangles

In this paper, we study Lebesgue differentiation processes along rectangles $R_k$ shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in $L^p$ spaces. In particular, classes of examples of such processes failing to converge a.e. in $L^{\mathrm{\infty }}$ 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 maximal operator associated to the set $\{k^{-s}:k\in {\mathbb{N}}^{\ast }\}$ fails to be bounded in $L^p$ for any $1\le p<\mathrm{\infty }$.