  The problem of characterizing which automatic sets of integers are stable is
here 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, a
necessary 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 and
Scanlon this gives a combinatorial characterization of the $d$-automatic
$A\subseteq \mathbb{Z}$ such that $(\mathbb{Z},+,A)$ is stable. This
characterization is in terms of what were called "$F$-sets" by Moosa and
Scanlon and "elementary $p$-nested sets" by Derksen. Automata-theoretic methods
are also used to produce some NIP expansions of $(\mathbb{Z},+)$, in particular
the expansion by the monoid $(d^\mathbb{N},\times )$.
