  Let $G$ be a commutative algebraic group embedded in projective space and
$\Gamma$ a finitely generated subgroup of $G$. From these data we construct a
chain of algebraic subgroups of $G$ which is intimately related to obstructions
to multiplicity or interpolation estimates. Let $\gamma_1,...,\gamma_l$ denote
a family of generators of $\Gamma$ and, for any $S>1$, let $\Gamma(S)$ be the
set of elements $n_1\gamma_1+..+n_l\gamma_l$ with integers $n_j$ such that
$|n_j| < S$. Then this chain of subgroups controls, for large values of $S$,
the distribution of $\Gamma(S)$ with respect to algebraic subgroups of $G$. As
an application we essentially determine (up to multiplicative constants) the
locus of common zeros of all $P \in H^0(\barG ,{\cal O}(D))$ which vanish to at
least some given order at all points of $\Gamma(S)$. When $D$ is very small
this result reduces to a multiplicity estimate; when $D$ is very large it is a
kind of interpolation estimate.
