Ratner/Masur equidistribution by orbit matching

In a 1992 article where she surveyed her recent breakthrough on unipotent flows on homogeneous spaces, Ratner presented an argument for the equidistribution of horospherical orbits in the context of horocycle flow on SL(2,R)/Lattice. This idea is separate from the ideas in her celebrated work on unipotent flows and I will present her argument for horospherical equidistribution in the simplest situation I can think of: proving the ergodicity of a particular directional flow on the flat two torus. Ratner's argument has similarities to Masur's criterion for unique ergodicity of translation flows, proven around the same time. Time permitting I will comment on Masur's criterion as well.