Oppenheim conjecture for pairs consisting of a linear form and a quadratic form

Gorodnik, Alexander

doi:10.1090/S0002-9947-04-03473-7

Oppenheim conjecture for pairs consisting of a linear form and a quadratic form
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Trans. Amer. Math. Soc. 356 (2004), 4447-4463 Request permission

Abstract:

Let $Q$ be a nondegenerate quadratic form and $L$ a nonzero linear form of dimension $d>3$. As a generalization of the Oppenheim conjecture, we prove that the set $\{(Q(x),L(x)):x\in \mathbb {Z}^d\}$ is dense in $\mathbb {R}^2$ provided that $Q$ and $L$ satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.

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