Consider families of spherical means where the radii are restricted to a given subset of a compact interval. One is interested in the L^p improving estimates for the associated maximal operators and related objects. Results depend on several notions of fractal dimension of the dilation set, or subsets of it. There are some unexpected statements on the shape of the possible type sets. Joint works with J. Roos, and with T. Anderson, K. Hughes and J. Roos.