In this talk we will consider spherical maximal operators in Euclidean space with a supremum taken over a given dilation set. It turns out that the sharp $L^p$ improving properties of such operators are closely related to fractal dimensions of the dilation set such as the Minkowski and Assouad dimensions.

At the center of the talk will be a simple characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of such a maximal operator.

This is joint work with Andreas Seeger. Time permitting, we will also discuss some ongoing work and further directions.