Joint work with Amir Mohammadi, Zhiren Wang, and Lei Yang

Let Q be an indefinite ternary quadratic form. In the 1980s Margulis proved the longstanding Oppenheim Conjecture, stating that unless Q is proportional to an integral form, the set of values Q attains at the integer points is dense in R. We give quantitative results to that effect.
In particular, if the coefficients of Q are algebraic, but Q is not proportional to an integral form, and if $(\alpha,\beta)$ a fixed interval, the number of integer points v in a ball of radius R for which $\alpha<Q(v)<\beta$ is asymptotic to $c(Q,\alpha,\beta)R$ as $R\to\infty$ with an effective power saving error term (but $c(Q,\alpha,\beta)$ might not be what you expect!).

Our work is based on a quantitative equidistribution result for unipotent flows, as well as upper bound estimates by Eskin-Margulis-Mozes and Wooyeon Kim.