Contributions to the theory of $F$-automatic sets

Abstract

Fix an abelian group Γ and an injective endomorphism F : Γ → Γ. Improving on the results of [2], 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 ≥ 7 that the expansion of (Fp[t], +) by multiplication restricted to t N is NIP.