Asymptotic density of rational sets in free abelian groups

TL;DR

This paper investigates the asymptotic density of rational sets within free abelian groups, providing methods to compute this density when the set is described via semi-simple decomposition.

Contribution

It establishes the existence of asymptotic density for rational sets in $\

Findings

Rational sets in $\

Method to compute asymptotic density from semi-simple decomposition

Abstract

In this paper we study asymptotic density of rational sets in free abelian group $\mathbb{Z}^n$ of rank $n$. We show that any rational set $R$ in $\mathbb{Z}^n$ has asymptotic density. If $R$ is given by its semi-simple decomposition we show how to compute its asymptotic density.