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

TL;DR

This paper proves that expanding translates of shrinking submanifolds in the space of unimodular lattices become equidistributed, leading to the conclusion that Dirichlet's approximation theorem cannot be improved for almost all points on certain submanifolds.

Contribution

It establishes the equidistribution of translated measures on submanifolds in the space of lattices, connecting dynamics with Diophantine approximation and confirming non-improvability results.

Findings

Translated measures become equidistributed as t→∞

Almost every point on certain submanifolds satisfies non-improvability of Dirichlet's theorem

Answers a longstanding question from Davenport and Schmidt (1969)

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).