Ratner's Theorems on Unipotent Flows and Their Applications to Quantitative Diophantine Approximation

This monograph presents a rigorous exposition of Ratner's Measure Classification Theorem and Orbit Closure Theorem within the framework of homogeneous dynamics. We analyze the ergodic properties of unipotent flows on the space of lattices SL(n, R)/SL(n, Z) and their profound implications for Number Theory. Specifically, we detail the derivation of the Oppenheim Conjecture and its quantitative refinements established by Eskin, Margulis, and Mozes. Furthermore, we explore the Dani Correspondence principle, which bridges the gap between the trajectory behavior of flows on homogeneous spaces and Diophantine properties of vectors in Rn. The discussion culminates in the Kleinbock-Margulis proof of Sprindzuk's Conjecture regarding the extremality of non-degenerate manifolds, utilizing quantitative non-divergence estimates.