Moments of Margulis functions and indefinite ternary quadratic forms

Abstract: In this paper, we study the moments of the Margulis $\alpha$-function integrating over expanding translates of a unipotent orbit in $\operatorname{SL}_3(\mathbb{R})/\operatorname{SL}_3(\mathbb{Z})$. We show that for some $\lambda>1$ the $\lambda$-moments of the Margulis $\alpha$-function over expanding translates of a unipotent orbit are uniformly bounded, under suitable Diophantine conditions of the initial unipotent orbit. As an application, we prove that for any indefinite irrational ternary quadratic form $Q$ with suitable Diophantine conditions and $a<b$ the number of integral vectors of norm at most $T$ satisfying $a<Q(v)<b$ is asymptotically equivalent to $\mathsf{C}_Q(b-a)T$ as $T$ tends to infinity, where the constant $\mathsf{C}_Q\>0$ depends only on $Q$.