Multiplicative structure in stable expansions of the group of integers

TL;DR

This paper explores the stability properties of certain expansions of the integers by unary predicates related to multiplicative structures and growth sequences, establishing super stability and $U$-rank results.

Contribution

It introduces two new families of expansions of $(Z,+,0)$ and proves their theories are superstable with $U$-rank $ ext{ extomega}$, advancing understanding of their model-theoretic complexity.

Findings

Theories of these expansions are superstable of $U$-rank extomega.

Stability is established for expansions by predicates of the form $ eq q^n$.

Identifies conditions under which sets grow asymptotically close to $Q$-independent sequences.

Abstract

We define two families of expansions of $(\mathbb{Z},+,0)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega$. The first family consists of expansions $(\mathbb{Z},+,0,A)$, where $A$ is an infinite subset of a finitely generated multiplicative submonoid of $\mathbb{N}$. Using this result, we also prove stability for the expansion of $(\mathbb{Z},+,0)$ by all unary predicates of the form $\{q^n:n\in\mathbb{N}\}$ for some $q\in\mathbb{N}_{\geq 2}$. The second family consists of sets $A\subseteq\mathbb{N}$ which grow asymptotically close to a $\mathbb{Q}$-linearly independent increasing sequence $(\lambda_n)_{n=0}^\infty\subseteq\mathbb{R}^+$ such that $\{\frac{\lambda_n}{\lambda_m}:m\leq n\}$ is closed and discrete.