Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

TL;DR

This paper demonstrates that under certain conditions, curves in matrix spaces are equidistributed in homogeneous spaces, leading to a generic form of Dirichlet's approximation theorem for matrices.

Contribution

It establishes a link between geometric properties of curves and their equidistribution in homogeneous spaces, extending Diophantine approximation results.

Findings

Almost every point on the curve cannot have improved Dirichlet approximation.

Translates of the curve become equidistributed in the homogeneous space under diagonal actions.

The geometric condition ensures the equidistribution property.

Abstract

In this paper, we study an analytic curve $\varphi: I=[a,b]\rightarrow \mathrm{M}(m\times n, \mathbb{R})$ in the space of $m$ by $n$ real matrices, and show that if $\varphi$ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into some homogeneous space $G/\Gamma$, and prove that under the action of some expanding diagonal subgroup $A= \{a(t): t \in \mathbb{R}\}$, the translates of the curve tend to be equidistributed in $G/\Gamma$, as $t \rightarrow +\infty$.