# Expansions of the real field by canonical products

## TL;DR

This paper investigates how expanding the real field with canonical Weierstrass products and related functions affects the structure's properties, revealing only three possible outcomes: o-minimality, d-minimality, or definability of integers.

## Contribution

It classifies the possible model-theoretic outcomes of expanding the real field with specific canonical products and associated functions, extending understanding of their definability properties.

## Key findings

- Only three outcomes: o-minimality, d-minimality, or definability of ta.
- Characterizes conditions under which each outcome occurs.
- Provides a framework for analyzing expansions by canonical Weierstrass products.

## Abstract

We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated to sequences such as $(-n^s)_{n>0}$ (for $s>0$) and $(-s^n)_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known for the resulting structures: (i)~o-minimality; (ii)~d-minimality (but not o-minimality); (iii)~definability of $\mathbb{Z}$.