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.Comment: 30 pages, 3 figures. Slightly improved Theorem 1.