Moments of Margulis functions and values of ternary quadratic forms

 The Oppenheim conjecture, proved by Margulis in 1986, states that for a non-degenerate indefinite irrational quadratic form Q in $n \geq 3$ 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. The quantitative Oppenheim conjecture was established by Eskin, Margulis, and Mozes for quadratic forms in $n \geq 4$ variables. In this talk, we discuss the quantitative Oppenheim conjecture for ternary quadratic forms (n=3). The main ingredient of the proof is a uniform boundedness result for the moments of Margulis functions over expanding translates of a unipotent orbit in the space of 3-dimensional lattices, under suitable Diophantine conditions of the initial unipotent orbit.