Equidistribution of expanding translates of curves and Diophantine approximation on matrices

TL;DR

This paper investigates how expanding translates of curves distribute in homogeneous spaces and applies the results to Diophantine approximation on matrices, showing almost all points on certain curves cannot improve Dirichlet's theorem.

Contribution

It establishes new equidistribution results for expanding translates of curves in homogeneous spaces and links these to Diophantine approximation properties on matrices.

Findings

Almost every point on certain analytic curves in matrix space cannot improve Dirichlet's theorem.

Identifies algebraic obstructions to equidistribution related to subvarieties of the flag variety.

Uses advanced techniques like Ratner's theorem and geometric invariant theory.

Abstract

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/\Gamma$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.