On a $\mathbb{Z}$-module connected to approximation theory

This paper deals with the set of $\alpha \in \mathbb{R}$ such that $\alpha \zeta^{n} \bmod 1$tendsto$0$forafixed$\zeta\in{$\mathbb{R}$}$, which we call ${\mathcal{M}}_{\zeta }$. Predominately the case of Pisot numbers $\zeta$ is studied. In this case the inclusions ${\mathcal{O}}_{\mathbb{Q}(\zeta )}\subset {\mathcal{M}}_{\zeta }\subset \mathbb{Q}(\zeta )$ are known. We will show the properties of ${\mathcal{M}}_{\zeta }$ are connected to the module structure of the ring of integers ${\mathcal{O}}_{\mathbb{Q}(\zeta )}$. We will describe the module structure of ${\mathcal{M}}_{\zeta }$ and how much ${\mathcal{M}}_{\zeta }$ differs from ${\mathcal{O}}_{\mathbb{Q}(\zeta )}$. The results besides allow to give some information on the shape of integral bases of real number fields.