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

Abstract

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).