Closures of locally divergent orbits of maximal tori and values of homogeneous forms

Let $\text{G}$ be a semisimple algebraic group over a number field $K$, $\mathcal{S}$ a finite set of places of $K$, $K_{\mathcal{S}}$ the direct product of the completions $K_v,v\in \mathcal{S}$, and $\text{OO}$ the ring of $\mathcal{S}$-integers of $K$. Let $G=\text{G}(K_{\mathcal{S}})$, $\mathrm{\Gamma }=\text{G}(\text{OO})$ and $\pi :G\to G\mathrm{/}\mathrm{\Gamma }$ the quotient map. We describe the closures of the locally divergent orbits $T\pi (g)$ %in $G\mathrm{/}\mathrm{\Gamma }$ where $T$ is a maximal $K_{\mathcal{S}}$-split torus in $G$. If $\mathrm{\#}S=2$ then the closure $\overline{T\pi (g)}$ is a finite union of $T$-orbits stratified in terms of parabolic subgroups of $\text{G}\times \text{G}$ and, consequently, $\overline{T\pi (g)}$ is homogeneous (i.e., $\overline{T\pi (g)}=H\pi (g)$ for a subgroup $H$ of $G$) if and only if $T\pi (g)$ is closed. On the other hand, if $# $\mathcal{S}$ > 2$and$K$isnota$$\mathrm{C}\mathrm{M}$$-fieldthen$\overline{T\pi(g)}$ is homogeneous for $\text{G}={\mathbf{S}\mathbf{L}}_n$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple $K$-ranks for $\text{G}\mathrm{\ne }{\mathbf{S}\mathbf{L}}_n$. As an application, we prove that $\overline{f({\text{OO}}^n)}=K_{\mathcal{S}}$ for the class of non-rational locally $K$-decomposable homogeneous forms $f\in K_{\mathcal{S}}[x_1,\cdots ,x_n]$.