Limit distributions of expanding translates of shrinking submanifolds and non-improvability of Dirichlet's approximation theorem

On the space $ {\mathscr{L}}_{n+1} $ of unimodular lattices in $ \mathbb{{R}}^{n+1} $, we consider the standard action of $ a(t) = {\rm{diag}}(t^n,t^{-1},\ldots,t^{-1})\in {\rm{SL}}(n+1, \mathbb{{R}}) $ for $ t>1 $. Let $ M $ be a nondegenerate submanifold of an expanding horospherical leaf in $ {\mathscr{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} $ gets equidistributed in $ {\mathscr{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 $ {\mathscr{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).