Intersections of diagonal orbits

TL;DR

This paper proves that every diagonal orbit in the space of lattices intersects both the set of stable lattices and the set of well-rounded lattices, revealing fundamental intersection properties in lattice dynamics.

Contribution

It establishes the universal intersection of diagonal orbits with both stable and well-rounded lattice sets in the space of lattices.

Findings

Any $A$-orbit intersects ${ m ST}_n$

Any $A$-orbit intersects ${ m WR}_n$

Results hold for all dimensions $n$

Abstract

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