Asymptotic distribution for pairs of linear and quadratic forms at integral vectors

We study the joint distribution of values of a pair consisting of a quadratic form ${\mathbf q}$ and a linear form ${\mathbf l}$ over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of $\mathrm{SL}_3(\mathbb{R})$ . Math. Ann.286 (1990), 101–128]. In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n \ge 5$ , then under the assumptions that for every $(\alpha , \beta ) \in {\mathbb {R}}^2 \setminus \{ (0,0) \}$ , the form $\alpha {\mathbf q} + \beta {\mathbf l}^2$ is irrational and that the signature of the restriction of ${\mathbf q}$ to the kernel of ${\mathbf l}$ is $(p, n-1-p)$ , where ${3\le p\le n-2}$ , the number of vectors $v \in {\mathbb {Z}}^n$ for which $\|v\| < T$ , $a < {\mathbf q}(v) < b$ and $c< {\mathbf l}(v) < d$ is asymptotically $ C({\mathbf q}, {\mathbf l})(d-c)(b-a)T^{n-3}$ as $T \to \infty $ , where $C({\mathbf q}, {\mathbf l})$ only depends on ${\mathbf q}$ and ${\mathbf l}$ . The density of the set of joint values of $({\mathbf q}, {\mathbf l})$ under the same assumptions is shown by Gorodnik [Oppenheim conjecture for pairs consisting of a linear form and a quadratic form. Trans. Amer. Math. Soc.356(11) (2004), 4447–4463].