Moments of Margulis functions and indefinite ternary quadratic forms

Abstract

In this paper, we prove a quantitative version of the Oppenheim conjecture for indefinite ternary quadratic forms: for any indefinite irrational ternary quadratic form $Q$ that is not extremely well approxiable by rational forms, and for $a<b$ the number of integral vectors of norm at most $T$ satisfying $a<Q(v)<b$ is asymptotically equivalent to $\big(\mathsf{C}_Q(b-a)+\mathsf{I}_{Q}(a,b)\big)T$ as $T$ tends to infinity, where the constant $\mathsf{C}_Q>0$ depends only on $Q$, and the term $\mathsf{I}_{Q}(a,b)T$ accounts for the contribution from rational isotropic lines and degenerate planes. The main technical ingredient is a uniform bound for the $\lambda$-moment of the Margulis $\alpha$-function along expanding translates of a unipotent orbit in $\operatorname{SL}_3(\mathbb{R})/\operatorname{SL}_3(\mathbb{Z})$, for some $\lambda>1$. To establish this, we introduce a new height function $\widetilde{\alpha}$ on the space of lattices, which captures the failure of the classical Margulis inequality. This moment bound implies equidistribution of such translates with respect to a class of unbounded test functions, including the Siegel transform.