On expansions of $(\mathbf{Z},+,0)$

Abstract

Call a (strictly increasing) sequence $(r_n)$ of natural numbers \emph{regular} if it satisfies the following condition: $r_{n+1}/r_n\to \theta \in {\mathbb{R}}^{>1}\cup \{\infty \}$ and, if $\theta$ is algebraic, then $(r_n)$ satisfies a linear recurrence relation whose characteristic polynomial is the minimal polynomial of $\theta$. Our main result states that $(\mathbb{Z},+,0,R)$ is superstable whenever $R$ is enumerated by a regular sequence. We give two proofs of this result. One relies on a result of E. Casanovas and M. Ziegler and the other on a quantifier elimination result. We also show that $(\mathbb{Z},+,0,<,R)$ is NIP whenever $R$ is enumerated by a regular sequence that is ultimately periodic modulo $m$ for all $m>1$.

33 pages; Final version. To appear in Ann. of Pure and Appl. Log

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