In the last three decades, the field of homogenous dynamics has proven to be an effective tool in studying asymptotic distribution problems in number theory. One of the landmark results in this area is the celebrated theorem of Eskin-Margulis-Mozes on the quantitative Oppenheim conjecture, which addresses the question of determining the asymptotic distribution of values of an irrational indefinite quadratic ${\mathbf q}$ over the set of integral vectors in a ball, as the radius goes to infinity.

In this talk we will discuss a generalization of this result involving the joint distribution of values of a pair consisting of a quadratic form and a linear form over the set of integral vectors, and elaborate on the tools used from homogenous dynamics that can be used to turn this counting problem into a question about equidistribution of certain shifted orbits in homogenous spaces.  This talk is based on the joint work with Jiyoung Han and Seonhee Lim.