Consider maximal operators for spherical means where the dilations are restricted to a given subset of a compact interval. We consider L^p improving estimates for such maximal functions. The outcomes relate to  various dimensional assumptions on the dilation set or its subsets. There are some surprising results on the shape of the  possible type set. 

This is joint work with Joris Roos and with Theresa Anderson, Kevin Hughes and Joris Roos.