This talk is about maximal averages on spheres in two and
higher-dimensional Euclidean space. This is a classic topic in harmonic analysis originating in questions
on differentiability properties of functions. We consider maximal spherical averages with a supremum taken over a
given dilation set. It turns out that the sharp Lp improving properties of such operators are closely related
to fractal dimensions of the dilation set such as the Minkowski and Assouad dimensions.
This leads to a surprising characterization of the closed convex sets which can occur as closure of the sharp Lp improving region of such a maximal operator. This is joint work with Andreas Seeger. If time allows we will also mention recent work on the Heisenberg group and related work in progress.