Oppenheim conjecture for pairs consisting of a linear form and a quadratic form
Alexander Gorodnik
Abstract
Let Q be a nondegenerate quadratic form, and L is 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∈ Zd is dense in R2 provided that Q and L satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.
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