New Effective Results Regarding the Oppenheim Conjecture and Polynomial Effective Equidistribution

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 (α,β)

an interval not containing 0, the number of integer points v in a sphere of radius R for which α<Q(v)<β

is asymptotic to the volume of the corresponding shape in R^3 with a power saving error term.

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