  Fix an abelian group $\Gamma$ and an injective endomorphism $F \colon \Gamma
\to \Gamma$. Improving on the results of Bell and Moosa, new characterizations
are here obtained for the existence of spanning sets, $F$-automaticity, and
$F$-sparsity. The model theoretic status of these sets is also investigated,
culminating with a combinatorial description of the $F$-sparse sets that are
stable in $(\Gamma, +)$, and a proof that the expansion of $(\Gamma, +)$ by any
$F$-sparse set is NIP. These methods are also used to show for prime $p\ge 7$
that the expansion of $(\mathbb{F}_p[t], +)$ by multiplication restricted to
$t^\mathbb{N}$ is NIP.
