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 q and a linear form l over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of SL₃(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 ≥ 5 , then under the assumptions that for every (α , β ) ∈ R² \ (0,0) \ , the form α q + β l² is irrational and that the signature of the restriction of q to the kernel of l is $(p, n-1-p)$ , where 3≤ p≤ n-2 , the number of vectors v ∈ Zⁿ for which $\|v\| < T$ , a < q(v) < b and c< l(v) < d is asymptotically C( q, l)(d-c)(b-a)Tⁿ⁻³ as T → ∞ , where C( q, l) only depends on q and l . The density of the set of joint values of ( q, 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].