Weakly minimal groups with a new predicate

TL;DR

This paper investigates the stability of expansions of weakly minimal structures by arbitrary subsets, especially focusing on pure abelian groups and providing new examples of stable and superstable structures through algebraic and recurrence-based subsets.

Contribution

It establishes that formulas in such expansions are equivalent to bounded formulas, characterizes stability via induced structures, and introduces new stable structures using algebraic and recurrence properties.

Findings

Formulas in expansions are equivalent to bounded formulas.

Stability depends on the induced structure on the subset.

Examples of superstable structures from algebraic groups and recurrence relations.

Abstract

Fix a weakly minimal (i.e., superstable $U$-rank $1$) structure $\mathcal{M}$. Let $\mathcal{M}^*$ be an expansion by constants for an elementary substructure, and let $A$ be an arbitrary subset of the universe $M$. We show that all formulas in the expansion $(\mathcal{M}^*,A)$ are equivalent to bounded formulas, and so $(\mathcal{M},A)$ is stable (or NIP) if and only if the $\mathcal{M}$-induced structure $A_{\mathcal{M}}$ on $A$ is stable (or NIP). We then restrict to the case that $\mathcal{M}$ is a pure abelian group with a weakly minimal theory, and $A_{\mathcal{M}}$ is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of $(\mathbb{Z},+)$. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form $(\mathcal{M},A)$. Most notably, we show that if $(G,+)$ is a weakly minimal additive subgroup of the algebraic numbers, $A\subseteq G$ is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of $A$ is a root of unity, then $(G,+,B)$ is superstable for any $B\subseteq A$.