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

Christopher D. C. Hawthorne

Published 2020-06-30Version 1

The problem of characterizing which automatic sets of integers are stable is here initiated. Given a positive integer $d$ and a subset $A\subseteq \mathbb{Z}^m$ whose set of representations base $d$ is sparse and 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 $d$-sparse $A\subseteq \mathbb{Z}^m$ 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. For $A\subseteq \mathbb{Z}$ $d$-automatic but not $d$-sparse, it is shown that if $x+y\in A$ is stable then finitely many translates of $A$ cover $\mathbb{Z}$. Automata-theoretic methods are also used to produce some NIP expansions of $(\mathbb{Z},+)$, in particular the expansion by the monoid $(d^\mathbb{N},\times )$.