This talk is about the inhomogeneous Diophantine approximations, that is, approximations of pairs (\Theta, \pmb{\eta}) of a matrix and a vector (or, equivalently, approximations of systems of affine forms) from the metric point of view. There are three essential ways to treat this setup: we can just look at all pairs (\Theta. \pmb{\eta}); we can fix the vector \pmb{\eta} and study the behavior of pairs for different \Theta, or we can fix some matrix \Theta. In recent years, a lot was done in the first two cases ("pairs" and "fixed vector"), however, many essential questions remained unanswered in the third ("fixed matrix").
We start with the classical transference principle to show how it gives answers to some of these questions. We will define the essential analogues for the classical sets of interest in Diophantine approximation, such as Badly approximable vectors and Dirichlet improvable vectors (now these notions will depend on the \Theta we fixed), and show that these sets behave in some sense similarly to the classical homogeneous analogues. In addition, we will show how our results provide an immediate and simple proof of Inhomogeneous Dirichlet's theorem by Kleinbock and Wadleigh.
