Certain actions of finitely generated groups on higher dimensional noncommutative tori

We study certain actions of finitely generated abelian groups on higher dimensional noncommutative tori. Given a dimension $d$ and a finitely generated abelian group $G$, we apply a certain function to detect whether there is a simple noncommutative $d$-torus which admits a specific action by $G$. We also describe the condition of $G$ under which the associated crossed product is an AF algebra.