Equidistribution of rational subspaces and their shapes

TL;DR

This paper proves the simultaneous equidistribution of rational subspaces and their associated lattice shapes in the Grassmannian, using unipotent dynamics under certain congruence conditions, except for the case (2,4).

Contribution

It introduces a new equidistribution result for rational subspaces and their lattice shapes, extending previous work with a focus on unipotent dynamics and congruence conditions.

Findings

Proves simultaneous equidistribution of subspaces and lattice shapes

Uses unipotent dynamics techniques

Results hold under specific congruence conditions

Abstract

To any $k$-dimensional subspace of $\mathbb Q^n$ one can naturally associate a point in the Grassmannian ${\rm Gr}_{n,k}(\mathbb R)$ and two shapes of lattices of rank $k$ and $n-k$ respectively. These lattices originate by intersecting the $k$-dimensional subspace with the lattice $\mathbb Z^n$. Using unipotent dynamics we prove simultaneous equidistribution of all of these objects under a congruence conditions when $(k,n) \neq (2,4)$.