Equidistribution of rational subspaces and their shapes

Abstract

To any $k$-dimensional subspace of ${\mathbb{Q}}^n$ one can naturally associate a point in the Grassmannian ${\mathrm{Gr}}_{n,k}(\mathbb{R})$ 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 $(k,n)\ne (2,4)$.

52 pages

References in corpus (6)

• Local-global principles for representations of quadratic forms
• Integer points and their orthogonal lattices
• Distribution of Shapes of orthogonal Lattices
• Integer points on spheres and their orthogonal grids
• Equidistribution of primitive lattices in ${\mathbb{R}}^n$
• On the distribution of primitive subgroups of ${\mathbb{Z}}^d$ of large covolume