Classifying orbits of the affine group over the integers

Abstract: For each $n=1,2,\dots$, let $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$ be the affine group over the integers. For every point $x=(x_1,\dots,x_n) \in \mathbb{R}^n$ let $\mathrm{orb}(x)={\gamma(x)\in \mathbb{R}^n\mid\gamma\in \mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n}.$ Let $G_{x}$ be the subgroup of the additive group $\mathbb R$ generated by $x_1,\dots,x_n, 1$. If $\mathrm{rank}(G_x)\neq n$ then $\mathrm{orb}(x)={y\in\mathbb{R}^n\mid G_y=G_x}$. Thus,$G_x$ is a complete classifier of $\mathrm{orb}(x)$. By contrast, if $\mathrm{rank}(G_x)=n$, knowledge of $G_x$ alone is not sufficient in general to uniquely recover $\mathrm{orb}(x)$: as a matter of fact, $G_x$ determines precisely $\mathrm{max}(1,\frac{\phi(d)}{2})$ different orbits, where $d$ is the denominator of the smallest positive nonzero rational in $G_x,$ and $\phi$ is Euler function. To get a complete classification, rational polyhedral geometry provides an integer $1\leq c_x\leq \mathrm{max}(1,d/2)$ such that $\mathrm{orb}(y)=\mathrm{orb}(x) $ iff $(G_{x},c_{x})=(G_{y},c_{{y}})$.