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For the spherical mean operators $\mathcal{A}_t$ in $\mathbb{R}^d$, $d\ge 2$,we consider the maximal functions $M_Ef =\sup_{t\in E} |\mathcal{A}_t f|$, withdilation sets $E\subset [1,2]$. In this paper we give a surprisingcharacterization of the closed convex sets which can occur as closure of thesharp $L^p$ improving region of $M_E$ for some $E$. This region depends on theMinkowski dimension of $E$, but also other properties of the fractal geometrysuch as the Assouad spectrum of $E$ and subsets of $E$. A key ingredient is anessentially sharp result on $M_E$ for a class of sets called (quasi-)Assouadregular which is new in two dimensions.

Spherical maximal functions and fractal dimensions of dilation sets