Paper detail

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

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