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

Abstract:

In this paper, 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. There is a natural map $u : \mathrm {M}(n\times n, \mathbb {R}) \rightarrow H = \mathrm {SL}(2n,\mathbb {R})$. Let $G$ be a Lie group containing $H$ and $\Gamma < G$ be a lattice of $G$. Let $X = G/\Gamma$. Then given a dense $H$-orbit in $X$, one could embed $u(\varphi (I))$ into $X$. We consider the expanding translates of the curve by some diagonal subgroup $A = \{a(t ) : t \in \mathbb {R}\} \subset H$. We will prove that if $\varphi$ satisfies certain geometric conditions, then the expanding translates will tend to be equidistributed in $G/\Gamma$, as $t \rightarrow +\infty$. As an application, we show that for almost every point on $\varphi (I)$, the Diophantine approximation given by Dirichlet’s Theorem is not improvable.