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The problem of characterizing which automatic sets of integers are stable ishere solved. Given a positive integer $d$ and a subset $A\subseteq \mathbb{Z}$whose set of representations base $d$ is recognized by a finite automaton, anecessary condition is found for $x+y\in A$ to be a stable formula in$\operatorname{Th}(\mathbb{Z},+,A)$. Combined with a theorem of Moosa andScanlon this gives a combinatorial characterization of the $d$-automatic$A\subseteq \mathbb{Z}$ such that $(\mathbb{Z},+,A)$ is stable. Thischaracterization is in terms of what were called "$F$-sets" by Moosa andScanlon and "elementary $p$-nested sets" by Derksen. Automata-theoretic methodsare also used to produce some NIP expansions of $(\mathbb{Z},+)$, in particularthe expansion by the monoid $(d^\mathbb{N},\times )$.

Automata and tame expansions of $(\mathbb{Z},+)$