A measure estimate in geometry of numbers and improvements to Dirichlet's theorem

Abstract

Let $ψ$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $ψ$-Dirichlet if for every sufficiently large real number $t$ one can find $\boldsymbol{p} \in \mathbb{Z}^m$, $\boldsymbol{q} \in \mathbb{Z}^n\smallsetminus\{\boldsymbol{0}\}$ satisfying $\|A\boldsymbol{q}-\boldsymbol{p}\|^m< ψ({t})$ and $\|\boldsymbol{q}\|^n<{t}$. This property was introduced by Kleinbock and Wadleigh in 2018, generalizing the property of $A$ being Dirichlet improvable which dates back to Davenport and Schmidt (1969). In the present paper, we give sufficient conditions on $ψ$ to ensure that the set of $ψ$-Dirichlet matrices has zero or full Lebesgue measure. Our proof is dynamical and relies on the effective equidistribution and doubly mixing of certain expanding horospheres in the space of lattices. Another main ingredient of our proof is an asymptotic measure estimate for certain compact neighborhoods of the critical locus (with respect to the supremum norm) in the space of lattices. Our method also works for the analogous weighted problem where the relevant supremum norms are replaced by certain weighted quasi-norms.