Differentiating Orlicz spaces with rare bases of rectangles

In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space. We also make a simple observation showing that the maximal operator associated to rectangles oriented in a fixed sequence of directions, is either bounded on all L p spaces for 1 < p , or fails to be bounded on any of them (adding the case p = to a dichotomy obtained previously by Bateman).