Equidistribution of expanding degenerate manifolds in the space of lattices

TL;DR

This paper establishes conditions under which expanding real-analytic submanifolds in the space of lattices become equidistributed under diagonal flows, extending previous results to degenerate cases and linking to Diophantine approximation.

Contribution

It provides necessary and sufficient conditions for equidistribution of degenerate manifolds in lattice spaces, generalizing Shah's non-degenerate case and connecting to Diophantine properties.

Findings

Necessary and sufficient conditions for equidistribution of degenerate manifolds.

Extension of Shah's results to degenerate submanifolds.

Almost every point on certain manifolds is not Dirichlet-improvable.

Abstract

For the space of unimodular lattices in a Euclidean space, we give necessary and sufficient conditions for equidistribution of expanding translates of any real-analytic submanifold under a diagonal flow. This extends the earlier result of Shah in the case of non-degenerate submanifolds. We apply the above dynamical result to show that if the affine span of a real-analytic submanifold in a Euclidean space satisfies certain Diophantine and arithmetic conditions, then almost every point on the manifold is not Dirichlet-improvable.