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

TL;DR

This paper establishes the asymptotic distribution of pairs of quadratic and linear forms evaluated at integral vectors, extending results related to the distribution of values of forms and their joint density.

Contribution

It provides a new asymptotic formula for the count of integral vectors with bounded norm and specified form value ranges, under certain irrationality and signature conditions.

Findings

Asymptotic count of vectors with bounded norm and form values derived

Conditions on forms ensure the distribution is well-behaved

Result generalizes previous distribution theorems for quadratic forms

Abstract

We study the joint distribution of values of a pair consisting of a quadratic form $q$ and a linear form $\mathbf l$ over the set of integral vectors, a problem initiated by Dani-Margulis (1989). 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 $\alpha q + \beta \mathbf l^2$ is irrational and that the signature of the restriction of $q$ 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 $\|v\| < T$, $a < q(v) < b$ and $c< \mathbf l(v) < d$ is asymptotically $$ C(q, \mathbf l)(d-c)(b-a)T^{n-3} , $$ as $T \to \infty$, where $C(q, \mathbf l)$ only depends on $q$ and $\mathbf l$. The density of the set of joint values of $(q, \mathbf l)$ under the same assumptions is shown by Gorodnik (2004).