Moments of Margulis functions and indefinite ternary quadratic forms

In this paper, we study the moments of the Margulis α-function integrating over expanding translates of a unipotent orbit in SL₃(R)/SL₃(Z). We show that for some λ>1 the λ-moments of the Margulis α-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 CQ(b-a)T as T tends to infinity, where the constant CQ>0 depends only on Q.