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 $A\mathrm{/}F$ be a simple abelian variety, $f:A\to {\mathbb{P}}^n$ be a morphism which is finite onto its image, and $\mathrm{\Gamma }\subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart ${\mathbb{A}}^n\subseteq {\mathbb{P}}^n$ and any finite subset $X\subseteq f(\mathrm{\Gamma })\cap {\mathbb{A}}^n$, the energy satisfies $E(X)\ll \mid X{\mid }^2$ and the sumset satisfies $\mid X+X\mid \gg \mid X{\mid }^2$. 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–Kühne.