## Illinois Journal of Mathematics

### Multiplicative structure in stable expansions of the group of integers

Gabriel Conant

#### Abstract

We define two families of expansions of $(\mathbb{Z},+)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega$. The first family consists of expansions $(\mathbb{Z},+,A)$, where $A$ is an infinite subset of a finitely generated multiplicative submonoid of $\mathbb{Z}^{+}$. Using this result, we also prove stability for the expansion of $(\mathbb{Z},+)$ 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.

#### Article information

Source
Illinois J. Math., Volume 62, Number 1-4 (2018), 341-364.

Dates
Received: 24 March 2018
Revised: 26 September 2018
First available in Project Euclid: 13 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1552442666

Digital Object Identifier
doi:10.1215/ijm/1552442666

Mathematical Reviews number (MathSciNet)
MR3922420

Zentralblatt MATH identifier
07036790

#### Citation

Conant, Gabriel. Multiplicative structure in stable expansions of the group of integers. Illinois J. Math. 62 (2018), no. 1-4, 341--364. doi:10.1215/ijm/1552442666. https://projecteuclid.org/euclid.ijm/1552442666

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