Counting and Joint equidistribution of approximates

Abstract

In this paper, we consider the problem of counting Diophantine inequalities with multiple natural constraints. We prove a very general result in this setting using dynamical techniques. More precisely, we consider the joint asymptotic distribution of $\varepsilon$-Diophantine approximates of matrices in several aspects. Our main results describe the resulting limiting measures for almost every matrix. Multiplicative Diophantine approximation is treated for the first time, and a number of Diophantine corollaries are derived. While we treat the general case of approximation of matrices, our results are already new for the case of simultaneous Diophantine approximation of vectors. Our approach is dynamical and is based on the construction of an appropriate Poincar\'{e} section for certain diagonal group actions on the space of unimodular lattices along with multiple mixing. The main new idea in our paper is a method that allows us to treat actions of higher rank groups.