The dynamics of subgroup actions on finite volume quotient spaces of Lie groups, also called homogeneous dynamics, is intimately connected to various questions on Diophantine approximation. In 2009, via proving a result on equidistribution of limits of expanding translates of analytic curves on the space of unimodular lattices in R^(n+1), Shah showed that for almost all points on any analytic curve on R^n which is not contained in a proper affine subspace, the Dirichlet's theorem on simultaneous approximation cannot be improved. Very recently, in a joint work with Shah, by proving a novel result on linear dynamics and equidistribution of expanding translates of shrinking curves, we have managed to generalize the non-improvablility result for points on regular smooth curves. At the heart of such results lies Ratner's theorem for unipotent flows. In this talk we will try to explain some of the concepts involved this circle of ideas.