Contributions to the theory of $F$-automatic sets
Hawthorne, Christopher · arXiv (Cornell University) · 2020
Fix an abelian group $Γ$ and an injective endomorphism $F \colon Γ\to Γ$. 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 $(Γ, +)$, and a proof that the expansion of $(Γ, +)$ 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.