Quantitative Equidistribution of Unipotent Flows on Homogeneous Spaces and the Effective Oppenheim Conjecture

This monograph explores the rigorous intersection of homogeneous dynamics and analytic number theory, focusing on the quantitative equidistribution of unipotent flows on quotients of semisimple Lie groups. We analyze the effective refinement of Ratner's equidistribution theorem, transitioning from the qualitative density results established by Margulis in his proof of the Oppenheim conjecture to precise asymptotic counting formulas with polynomial error terms. Central to this exposition is the work of Eskin, Margulis, and Mozes, which utilizes the linearization method and Margulis functions to control excursions into the cusp of non-compact homogeneous spaces. We further discuss recent advances by Lindenstrauss, Mohammadi, and Wang regarding polynomial effective rates for one-parameter unipotent subgroups. The derivation of the effective Oppenheim conjecture is presented as a corollary of these dynamical theorems, establishing lower bounds on the exponents of error terms for the distribution of values of indefinite irrational quadratic forms at integer points.