The theory of Diophantine approximation is underpinned by Dirichlet’s fundamental theorem. Broadly speaking, the main questions in the theory concern quantifying the prevalence of points with exceptional behavior with respect to Dirichlet’s result. Badly approximable, very well approximable and Dirichlet-improvable points are among the most well-studied such exceptional sets. The work of Dani and Kleinbock-Margulis connects these questions to the recurrence behavior of certain flows on homogeneous spaces. After a brief overview of the subject and the motivating questions, I will discuss new results giving a sharp upper bound on the Hausdorff dimension of divergent orbits of certain diagonal flows emanating from fractals on the space of lattices. Connections to theory of projections of self-similar measures will be presented.