Singular matrices and geometry at infinity of products of symmetric spaces

Abstract

Let $\mathbf{k}$ be a number field and $\mathbf{k}_M$ the Minkowski space associated to $\mathbf{k}$. Dirichlet's theorem in Diophantine approximation is generalized to the case of systems of linear forms with coefficients in $\mathbf{k}_M$. We study the set of singular systems in this setting. We generalize the transference principle, Dani's correspondence and give an estimate of the Hausdorff dimension of the set of singular systems from below.