Directional Statistics of Lattice Points and Escape of Mass for Embedded Horospheres

I will discuss escape of mass estimates for SL(d,ℝ)-horospheres embedded in the space of affine lattices, which depend on the Diophantine properties of the shortest affine lattice vector. These estimates can be used, in conjunction with Ratner's theorem, to prove the convergence of moments in natural lattice point problems, including the statistics of directions in lattices, inhomogeneous Farey factions, and the distribution of smallest denominators.

Based on joint work with Wooyeon Kim (ETH).