Metric theory of inhomogeneous Diophantine approximations with a fixed matrix

In this paper we develop a metric theory of inhomogeneous Diophantine approximation for the case of a fixed matrix. We use transference principle to connect uniform Diophantine properties of a pair $(\Theta, \pmb{\eta})$ of a matrix and a vector with the asymptotic Diophantine properties of the transposed matrix $\Theta^{\top}$, and vice versa, the asymptotic Diophantine properties of a pair $(\Theta, \pmb{\eta})$ with asymptotic Diophantine properties of the transposed matrix. In these setups, we prove analogues of classical statements of metrical homogeneous Diophantine approximations and answer some open questions that were raised in recent works.