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

This paper deals with the set of $\alpha\in{\mathbb{R}}$ such that $\alpha \zeta^{n} \bmod 1$ tends to $0$ for a fixed $\zeta\in{\mathbb{R}}$, which we call $\mathscr{M}_{\zeta}$. Predominately the case of Pisot numbers $\zeta$ is studied. In this case the inclusions $\mathcal{O}_{\mathbb{Q}(\zeta)}\subset\mathscr{M}_{\zeta}\subset\mathbb{Q}(\zeta)$ are known. We will show the properties of $\mathscr{M}_{\zeta}$ are connected to the module structure of the ring of integers $\mathcal{O}_{\mathbb{Q}(\zeta)}$. We will describe the module structure of $\mathscr{M}_{\zeta}$ and how much $\mathscr{M}_{\zeta}$ 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.