Additive Rigidity for Images of Rational Points on Abelian Varieties I: The Simple Case

We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to projective space. Let $A/F$ be a simple abelian variety, $f:A \rightarrow \mathbb{P}^n$ be a morphism which is finite onto its image, and $\Gamma \subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart $\mathbb{A}^n \subseteq \mathbb{P}^n$ and any finite subset $X \subseteq f(\Gamma) \cap \mathbb{A}^n$, the energy satisfies $E(X) \ll \lvert X \rvert^2$ and the sumset satisfies $\lvert X+X \rvert \gg \lvert X \rvert^2$. We then ask whether the same additive rigidity holds for arbitrary abelian varieties, and prove that this is indeed the case when the morphism $f$ is compatible with the decomposition of $A$ into simple factors. The proof uses the uniform Mordell-Lang conjecture proven by Gao--Ge--K\"{u}hne.