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Asymptotic distribution for pairs of linear and quadratic forms at integral vectors
Published online by Cambridge University Press: 17 April 2024
Abstract
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].
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