  This work is the third part of a series of papers. In the first two we
consider curves and varieties in a power of an elliptic curve. Here we deal
with subvarieties of an abelian variety in general.
  Let V be an irreducible variety of dimension d embedded in an abelian variety
A, both defined over the algebraic numbers. We say that V is weak-transverse if
V is not contained in any proper algebraic subgroup of A, and transverse if it
is not contained in any translate of such a subgroup.
  Assume a conjectural lower bound for the normalized height of V. For V
transverse, we prove that the algebraic points of bounded height of V which lie
in the union of all algebraic subgroups of A of codimension at least d+1
translated by the points close to a subgroup G of finite rank are non
Zariski-dense in V. If G has rank zero, it is sufficient to assume that V is
weak-transverse.
  The notion of closeness is defined using a height function.
