Homogeneous orbit closures and applications

TL;DR

This paper constructs new examples of orbit closures in the space of lattices, demonstrating both homogeneous and irregular cases, with applications to Diophantine approximation involving cubic roots of 2.

Contribution

It introduces new classes of orbit closures for diagonal group actions on lattices, including explicit irregular examples and applications to number theory.

Findings

New homogeneous orbit closures identified

Explicit irregular orbit closures constructed

Diophantine approximation results for cubic roots of 2

Abstract

We give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in R^d for d > 2 with nice (homogeneous) orbit closures, as well as examples of orbits with explicitly computable but irregular orbit closures. We give Diophantine applications to the former, for instance we show that if x is the cubic root of 2 then for any y,z in R liminf |n|<nx-y><nx^2-z>=0 (as |n| goes to infinity), where <c> denotes the distance of a real number c to the integers.