Multi-dimensional metric approximation by primitive points

Abstract

We consider diophantine inequalities of the form $| {\Theta }\mathbf{q}+ \mathbf{p}- \mathbf{y}|\le \psi (| \mathbf{q}|)$, with $\Theta \in \mathrm{Mat}_{n,m}({\mathbb R})$, $\mathbf{y}\in {\mathbb R}^n$, where $m,n\in {\mathbb N}$, and $\psi$ is a function on ${\mathbb N}$ with positive real values, seeking integral solutions $\mathbf{q}\in {\mathbb Z}^m$ and $\mathbf{p}\in {\mathbb Z}^n$ for which the restriction of the vector $(\mathbf{q}, \mathbf{p})^t$ to the components of a given partition $\pi$ are primitive integer points. In this setting, we establish metrical statements in the style of the Khintchine–Groshev Theorem. Similar solutions are considered for the doubly metrical inequality $| {\Theta }\mathbf{q}+\Phi \mathbf{p}- \mathbf{y}|\le \psi (| \mathbf{q}|)$, with $\Phi \in \mathrm{Mat}_{n,n}({\mathbb R})$ (other notations as before). The results involve the conditions that $x \mapsto x^{m-1}\psi (x)^n$ be non-increasing, and that the components of $\pi$ have at least $n+1$ elements each.