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

TL;DR

This paper establishes measure-theoretic criteria for the set of matrices satisfying a generalized Dirichlet approximation property, using dynamical systems techniques involving equidistribution and mixing in the space of lattices.

Contribution

It provides new sufficient conditions on the approximation function  for the measure of -Dirichlet matrices to be zero or full, extending previous work with a dynamical approach.

Findings

Criteria for zero measure of -Dirichlet matrices.

Criteria for full measure of -Dirichlet matrices.

Extension to weighted quasi-norms in approximation.

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 $\boldsymbol{p} \in \mathbb{Z}^m$, $\boldsymbol{q} \in \mathbb{Z}^n\smallsetminus\{\boldsymbol{0}\}$ satisfying $\|A\boldsymbol{q}-\boldsymbol{p}\|^m< \psi({t})$ 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 $\psi$ to ensure that the set of $\psi$-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.