Multi-dimensional metric approximation by primitive points

We consider diophantine inequalities of the form , with , , where , and is a function on with positive real values, seeking integral solutions and for which the restriction of the vector to the components of a given partition 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 , with (other notations as before). The results involve the conditions that be non-increasing, and that the components of have at least elements each.