We consider diophantine inequalities of the form |Θq+p-y|≤ψ(|q|)\documentclass[12pt]{minimal}
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\begin{document}$$| {\Theta }\mathbf{q}+ \mathbf{p}- \mathbf{y}|\le \psi (| \mathbf{q}|)$$\end{document}, with Θ∈Matn,m(R)\documentclass[12pt]{minimal}
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\begin{document}$$\Theta \in \mathrm{Mat}_{n,m}({\mathbb R})$$\end{document}, y∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{y}\in {\mathbb R}^n$$\end{document}, where m,n∈N\documentclass[12pt]{minimal}
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\begin{document}$$m,n\in {\mathbb N}$$\end{document}, and ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} is a function on N\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb N}$$\end{document} with positive real values, seeking integral solutions q∈Zm\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{q}\in {\mathbb Z}^m$$\end{document} and p∈Zn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{p}\in {\mathbb Z}^n$$\end{document} for which the restriction of the vector (q,p)t\documentclass[12pt]{minimal}
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\begin{document}$$(\mathbf{q}, \mathbf{p})^t$$\end{document} to the components of a given partition π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} 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 |Θq+Φp-y|≤ψ(|q|)\documentclass[12pt]{minimal}
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\begin{document}$$| {\Theta }\mathbf{q}+\Phi \mathbf{p}- \mathbf{y}|\le \psi (| \mathbf{q}|)$$\end{document}, with Φ∈Matn,n(R)\documentclass[12pt]{minimal}
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\begin{document}$$\Phi \in \mathrm{Mat}_{n,n}({\mathbb R})$$\end{document} (other notations as before). The results involve the conditions that x↦xm-1ψ(x)n\documentclass[12pt]{minimal}
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\begin{document}$$x \mapsto x^{m-1}\psi (x)^n$$\end{document} be non-increasing, and that the components of π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} have at least n+1\documentclass[12pt]{minimal}
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\begin{document}$$n+1$$\end{document} elements each.
