Locally divergent orbits of maximal tori and values of forms at integral points

Abstract: Let $\G$ be a semisimple algebraic group defined over a number field $K$, $\te$ a maximal $K$-split torus of $\G$, $\mathcal{S}$ a finite set of valuations of $K$ containing the archimedean ones, $\OO$ the ring of $\mathcal{S}$-integers of $K$ and $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$. Denote $G = \G(K_\mathcal{S})$, $T = \te(K_\mathcal{S})$ and $\Gamma = \G(\OO)$. Let $T\pi(g)$ be a locally divergent orbit for the action of $T$ on $G/\Gamma$ by left translations. We prove: ($1$) if $# S = 2$ then the closure $\overline{T\pi(g)}$ is a union of finitely many $T$-orbits all stratified in terms of parabolic subgroups of $\G \times \G$ and, therefore, $\overline{T\pi(g)}$ is homogeneous only if ${T\pi(g)}$ is closed, ($2$) if $# \mathcal{S} > 2$ and $K$ is not a $\mathrm{CM}$-field then $\overline{T\pi(g)}$ is squeezed between closed orbits of two reductive groups of equal semisimple ranks implying that $\overline{T\pi(g)}$ is homogeneous when $\G = \mathbf{SL}{n}$. As an application, if $f = (f_v){v \in \mathcal{S}} \in K_{\mathcal{S}}[x_1, \cdots, x_{n}]$, where $f_v$ are non-pairwise proportional decomposable over $K$ homogeneous forms, then $f(\OO^{n})$ is dense in $K_{\mathcal{S}}$.