Intersections of diagonal orbits

Abstract

Let $A\subseteq S{L}_n(\mathbb{R})$ group of diagonal matrices with positive diagonal, let ${\mathrm{ST}}_n\subseteq X_n:=S{L}_n(\mathbb{R})/S{L}_n(\mathbb{Z})$ be the set of stable lattices, and let ${\mathrm{WR}}_n\subseteq X_n$ be the set of well-rounded lattices. We prove that any $A$-orbit in $X_n$ intersects both ${\mathrm{ST}}_n$ and ${\mathrm{WR}}_n$.