Structure theory of set addition with two operations

We take the first step toward a structure theory that includes both operations of a ring $\mathcal{R}$. More precisely, we prove a series of inverse results for the structure of sets $A\subseteq {\mathbf{F}}_p$ such that, under certain conditions on integers $r_1,\dots ,r_k$, one has $\mathrm{\mid }{A}^{r_1}+\cdots +A^{r_k}\mathrm{\mid }\ll \sqrt[k]{p^{k-1}\mathrm{\mid }A\mathrm{\mid }}$.