  On the space $\mathcal{L}_{n+1}$ of unimodular lattices in
$\mathbb{R}^{n+1}$, we consider the standard action of
$a(t)=\mathrm{diag}(t^n,t^{-1},\ldots,t^{-1})\in \mathrm{SL}(n+1,\mathbb{R})$
for $t>1$. Let $M$ be a nondegenerate submanifold of an expanding horospherical
leaf in $\mathcal{L}_{n+1}$. We prove that for all $x\in M\setminus E$ and
$t>1$, if $\mu_{x,t}$ denotes the normalized Lebesgue measure on the ball of
radius $t^{-1}$ around $x$ in $M$, then the translated measure $a(t)\mu_{x,t}$
get equidistributed $\mathcal{L}_{n+1}$ as $t\to\infty$, where $E$ is a union
of countably many lower dimensional submanifolds of $M$. In particular, if
$\mu$ is an absolutely continuous probability measure on $M$, then $a(t)\mu$
gets equidistributed in $\mathcal{L}_{n+1}$ as $t\to\infty$. This result
implies the non-improvability of Dirichlet's Diophantine approximation theorem
for almost every point on a $C^{n+1}$-submanifold of $\mathbb{R}^n$ satisfying
a non-degeneracy condition, answering a question arising from the work of
Davenport and Schmidt (1969).
