An exposition on the supersimplicity of certain expansions of the additive group of the integers

TL;DR

This paper provides a clear exposition of the supersimplicity property in specific expansions of the additive group of integers, including predicates for special sets like primes and square-free numbers.

Contribution

It offers a self-contained explanation of supersimplicity in these expansions, referencing recent results and conjectures in model theory.

Findings

Supersimplicity holds for expansions with a generic predicate.

Supersimplicity holds for expansions with a predicate for square-free integers.

Supersimplicity holds for expansions with a predicate for prime integers, assuming Dickson's conjecture.

Abstract

In this short note, we present a self-contained exposition of the supersimplicity of certain expansions of the additive group of the integers, such as adding a generic predicate (due to Chatzidakis and Pillay), a predicate for the square-free integers (due to Bhardwaj and Tran) or a predicate for the prime integers (due to Kaplan and Shelah, assuming Dickson's conjecture).