Dirichlet's theorem on diophantine approximation and homogeneous flows

Abstract

We show that for any $ε<1$ and any $\mathcal{T}$ `drifting away from walls&#39;, Dirichlet&#39;s Theorem cannot be $ε$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions). In the case $m = 1$ we also show that for a large class of measures $μ$ there is $ε_0>0$ such that for any drifting away from walls $\mathcal{T}$, any $ε<ε_0$, and for $μ$-almost every $Y$, Dirichlet&#39;s Theorem cannot be $ε$-improved along $\mathcal{T}$. These measures include natural measures on sufficiently regular smooth manifolds and fractals. Our results extend those of several authors beginning with the work of Davenport and Schmidt done in late 1960s. The proofs rely on a translation of the problem into a dynamical one regarding the action of a diagonal semigroup on the space $\text{SL}_{m+n}(\mathbb{R})/\text{SL}_{m+n}(\mathbb{Z})$.