Consider maximal operators for spherical means where the dilations are restricted to a given subset of a compact interval. We discuss \(L^p\) improving estimates for such operators and connections to related global problems. The results depend on various notions of fractal dimensions of the dilation set or subsets of it. There are some surprising results on the shape of the possible type set. This is joint work with Joris Roos, and also relates to earlier work with Theresa Anderson, Kevin Hughes and Joris Roos.