Additive Rigidity for Images of Rational Points on Abelian Varieties

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.