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

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 $\Vert v\Vert <T$, $a<q(v)<b$ 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).

23 pages

References in corpus (1)

• Weight decomposition of ${\mathfrak{sl}}_d(\mathbb{R})$ with respect to the adjoint representation of $\mathfrak{so}(p,q)$