Title: Distribution of the values of ternary quadratic forms
Abstract: The Oppenheim conjecture, proved by Margulis in 1986, states that for a non-degenerate indefinite irrational quadratic form Q in n≥ 3 variables, the image set Q(Z^n) of integral vectors is a dense subset of the real line. Determining the distribution of values of an indefinite quadratic form at integral points asymptotically is referred to as quantitative Oppenheim conjecture. This problem was resolved by Eskin, Margulis, and Mozes for quadratic forms in n≥ 4 variables.
In this talk, we discuss the case of ternary quadratic forms (n=3). The approach uses ideas from homogeneous dynamics, where the problem is translated into an equidistribution problem for certain unipotent flows on the space of 3-dimensional lattices. The main ingredient is a strong quantitative non-divergence estimate, which controls how long these unipotent flows can spend deep in the cusp of the space of lattices.