Hausdorff dimension estimates for bounded orbits on homogeneous spaces of Lie groups

This work is motivated by studying badly approximable vectors, that is, ${\bf x}\in {\bf R}^n$ such that $\|q{\bf x} - {\bf p}\| \ge cq^{-1/n}$ for all ${\bf p}\in {\bf Z}^n$, $q\in {\bf N}$. Computing the Hausdorff dimension of the set of such ${\bf x}$ for fixed $c$ is an open problem. I will present some estimates, based on the interpretation of a badly approximable vector via a trajectory on the space of lattices, and then use exponential mixing to estimate from above the dimension of points whose trajectories stay in a fixed compact set. Joint work with Ryan Broderick.