Spherical maximal functions and fractal dimensions of dilation sets

For the spherical mean operators At in Rd, d≥2, we consider the maximal functions MEf=supt∈E|Atf|, with dilation sets E⊂[1,2]. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp Lp improving region of ME for some E. This region depends on the Minkowski dimension of E, but also other properties of the fractal geometry such the Assouad spectrum of E and subsets of E. A key ingredient is an essentially sharp result on ME for a class of sets called (quasi-)Assouad regular which is new in two dimensions.