Stable groups and expansions of $(\mathbb{Z},+,0)$

TL;DR

The paper proves that certain stable groups are superstable with finite U-rank and shows that the group of integers has no proper stable expansions of finite weight, answering a question about stable expansions in Presburger arithmetic.

Contribution

It establishes conditions under which stable groups are superstable of finite U-rank and demonstrates that the integers have no nontrivial stable expansions of finite weight.

Findings

Stable groups of finite weight with no infinite chains are superstable of finite U-rank.

The group of integers $( abla, +, 0)$ admits no proper stable expansions of finite weight.

Stable expansions of $( abla, +, 0)$ by sets definable in finite dp-rank expansions are already definable in $( abla, +, 0)$.

Abstract

We show that if $G$ is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then $G$ is superstable of finite $U$-rank. Combined with recent work of Palacin and Sklinos, we conclude that $(\mathbb{Z},+,0)$ has no proper stable expansions of finite weight. A corollary of this result is that if $P\subseteq\mathbb{Z}^n$ is definable in a finite dp-rank expansion of $(\mathbb{Z},+,0)$, and $(\mathbb{Z},+,0,P)$ is stable, then $P$ is definable in $(\mathbb{Z},+,0)$. In particular, this answers a question of Marker on stable expansions of the group of integers by sets definable in Presburger arithmetic.