Spherical maximal functions and fractal dimensions of dilation sets

For the spherical mean operators ${\mathcal{A}}_t$ in ${\mathbb{R}}^d$, $d\ge 2$, we consider the maximal functions $M_{E}f={\sup }_{t\in E}\mathrm{\mid }{\mathcal{A}}_{t}f\mathrm{\mid }$, with dilation sets $E\subset [1,2]$. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of $M_E$ for some $E$. This region depends on the Minkowski dimension of $E$, but also other properties of the fractal geometry such as the Assouad spectrum of $E$ and subsets of $E$. A key ingredient is an essentially sharp result on $M_E$ for a class of sets called (quasi-)Assouad regular which is new in two dimensions.