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

Abstract

Let $\psi$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $\psi$-Dirichlet if for every sufficiently large real number $t$ one can find $\mathbf{p} \in \mathbb{Z}^m$, $\mathbf{q} \in \mathbb{Z}^n\setminus\{\mathbf{0}\}$ satisfying $\|A\mathbf{q}-\mathbf{p}\|^m< \psi(t)$ and $\|\mathbf{q}\|^n<t$. By removing a technical condition from a partial zero-one law proved by Kleinbock-Str\"ombergsson-Yu, we prove a zero-one law for the Lebesgue measure of the set of $\psi$-Dirichlet matrices provided that $\psi(t)<1/t$ and $t\psi(t)$ is increasing. In fact, we prove the zero-one law in a more general situation with the monotonicity assumption on $t\psi(t)$ replaced by a weaker condition. Our proof follows the dynamical approach of Kleinbock-Str\"ombergsson-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-Str\"ombergsson-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.