Equidistribution of expanding curves in homogeneous spaces and Diophantine approximation for square matrices

TL;DR

This paper demonstrates that under certain conditions, expanding curves in the space of real matrices become equidistributed in a homogeneous space, leading to almost sure non-improvability of Diophantine approximation for typical points.

Contribution

It establishes equidistribution of expanding curves in homogeneous spaces and applies this to Diophantine approximation for matrix spaces, addressing a specific case of a conjecture by Nimish Shah.

Findings

Expanding curves tend to be equidistributed in homogeneous spaces.

Almost every point on the curve exhibits non-improvable Diophantine approximation.

The result solves a special case of a problem posed by Nimish Shah.

Abstract

In this article, we study an analytic curve $\varphi: I=[a,b]\rightarrow \mathrm{M}(n\times n, \mathbb{R})$ in the space of $n$ by $n$ real matrices, and show that if $\varphi$ satisfies certain geometric conditions, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem is not improvable. To do this, we embed the curve into some homogeneous space $G/\Gamma$, and prove that under the action of some expanding diagonal flow $A= \{a(t): t \in \mathbb{R}\}$, the expanding curves tend to be equidistributed in $G/\Gamma$, as $t \rightarrow +\infty$. This solves a special case of a problem proposed by Nimish Shah in ~\cite{Shah_1}.