A zero-one law for improvements to Dirichlet's theorem in arbitrary dimension

Abstract

Let $\ddot{I}ˆ$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $\ddot{I}ˆ$-Dirichlet if for every sufficiently large real number $t$ one can find $\mathbf{p}\in {\mathbb{Z}}^m$, $\mathbf{q}\in {\mathbb{Z}}^n\setminus \{0\}$ satisfying $\Vert A\mathbf{q}-\mathbf{p}{\Vert }^m<\ddot{I}ˆ(t)$ and $\Vert \mathbf{q}{\Vert }^n<t$. By removing a technical condition from a partial zero-one law proved by Kleinbock-Strömbergsson-Yu, we prove a zero-one law for the Lebesgue measure of the set of $\ddot{I}ˆ$-Dirichlet matrices provided that $\ddot{I}ˆ(t)<1/t$ and $t\ddot{I}ˆ(t)$ is increasing. In fact, we prove the zero-one law in a more general situation with the monotonicity assumption on $t\ddot{I}ˆ(t)$ replaced by a weaker condition. Our proof follows the dynamical approach of Kleinbock-Strömbergsson-Yu in reducing the question to a shrinking target problem in the space of lattices. The key new ingredient is a family of carefully chosen subsets of the shrinking targets studied by Kleinbock-Strömbergsson-Yu, together with a short-range mixing estimate for the associated hitting events. Our method also works for the analogous weighted problem where the relevant supremum norms are replaced by certain weighted quasi-norms.

22 pages