An "almost all versus no" dichotomy for orbits on homogeneous spaces
Several years ago I proved (in an involved and somewhat mysterious way) the following theorem:
suppose M is an analytic submanifold of R^n which contains a not very well approximable
vector; then almost all its vectors are not very well approximable. Recently I found a
simple argument establishing this and other more general results. Both old and new proofs
use quantitative nondivergence on the space of lattices. No background is necessary to follow
the talk, everything will be explained from scratch.