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.
Categories: math.LO