  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.
