In this article, we study an analytic curve $arphi: I=[a,b]ightarrowmathrm{M}(nimes n, mathbb{R})$ in the space of $n$ by $n$ real matrices,and show that if $arphi$ satisfies certain geometric conditions, then foralmost every point on the curve, the Diophantine approximation given byDirichlet's Theorem is not improvable. To do this, we embed the curve into somehomogeneous space $G/Gamma$, and prove that under the action of some expandingdiagonal flow $A= {a(t): t in mathbb{R}}$, the expanding curves tend to beequidistributed in $G/Gamma$, as $t ightarrow +infty$. This solves aspecial case of a problem proposed by Nimish Shah in ~cite{Shah_1}.
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机译：在本文中，我们研究分析曲线$ varphi：I = [a，b] rightarrow mathrm {M}（n times n， mathbb {R}）$$$ n乘以$ n $实数矩阵，并表明如果$ varphi $满足某些几何条件，那么几乎曲线上的每个点，Dirichlet定理给出的Diophantine逼近都是不可改进的。为此，我们将曲线嵌入到均匀的空间$ G / Gamma $中，并证明在某些对角线扩展$ A = {a（t）：t in mathbb {R} } $的作用下，扩展曲线倾向于以$ G / Gamma $进行等额分配，如$ t rightarrow + infty $。这解决了Nimish Shah在〜 cite {Shah_1}中提出的问题的特殊情况。
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