The Mordell–Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety X with a subgroup of finite rank is contained in a finite union of cosets contained in X . In this article, we prove a uniform version of this conjecture, meaning that the number of cosets necessary depend only on the dimension of the ambient variety and the degree (with respect to some polarization) of the subvariety X . To achieve this, we prove a general gap principle on algebraic points that extends the new gap principle for curves embedded into their Jacobians, previously obtained by Dimitrov–Gao–Habegger and Kühne. Our new gap principle also implies the full uniform Bogomolov conjecture in abelian varieties.
Gao et al. (Wed,) studied this question.