  Let $S$ be a finitely generated abelian semigroup of invertible linear
operators on a finite dimensional real or complex vector space $V$. We show
that every coarsely dense orbit of $S$ is actually dense in $V$. More
generally, if the orbit contains a coarsely dense subset of some open cone $C$
in $V$ then the closure of the orbit contains the closure of $C$. In the
complex case the orbit is then actually dense in $V$. For the real case we give
precise information about the possible cases for the closure of the orbit.
