For an $n$-by-$m$ matrix $X$ and a point $y\in\mathbb{R}^m$, we seek solutions $(\mathbf{p},\mathbf{q})\in \mathbb{Z}^m\times\mathbb{Z}^n$ to the inequality $|\mathbf{q}X - \mathbf{p} - y| < \psi(|q|)$, where $\psi$ is some fixed function of the natural numbers. Following a framework introduced by Dani, Laurent, and Nogueira in 2015, we impose additional coprimality-type conditions on the entries in the $(n+m)$-vector $(\mathbf{p}, \mathbf{q})$. In this set up, Dani, Laurent, and Nogueira proved results analogous to a doubly-metric inhomogeneous Khintchine--Groshev theorem, and they asked: 1) whether the result could be made singly-metric; 2) whether the family of "coprimality conditions" they considered could be expanded; 3) whether a monotonicity assumption could be removed from their results. I will discuss this as well as recent work with Demi Allen where we address the three questions.
