The definitive work of Margulis, settling the conjecture in the affirmative, used methods arising from ergodic theory and the study of discrete subgroups of semisimple Lie groups.
Meyer's theorem states that an indefinite integral quadratic form Q in n variables, n ≥ 5, nontrivially represents zero, i.e. there exists a non-zero vector x with integer components such that Q(x) = 0.
Geometry of actions of certain unipotent subgroups of the orthogonal group on the homogeneous space of the lattices in R3 plays a decisive role in this approach.
The idea to derive the Oppenheim conjecture from a statement about homogeneous group actions is usually attributed to M. S. Raghunathan, who observed in the 1970s that the conjecture for n = 3 is equivalent to the following property of the space of lattices: However, Margulis later remarked that in an implicit form this equivalence occurred already in a 1955 paper of Cassels and H. P. F. Swinnerton-Dyer, albeit in a different language.
The study of the properties of unipotent and quasiunipotent flows on homogeneous spaces remains an active area of research, with applications to further questions in the theory of Diophantine approximation.