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Transactions of the American Mathematical Society

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

Author: Alexander Gorodnik
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4447-4463
MSC (2000): Primary 11J13, 11H55, 37A17
Published electronically: January 13, 2004
MathSciNet review: 2067128
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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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Additional Information

Alexander Gorodnik
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Received by editor(s): November 29, 2002
Received by editor(s) in revised form: May 9, 2003
Published electronically: January 13, 2004
Additional Notes: This article is a part of the author’s Ph.D. thesis at Ohio State University done under the supervision of Professor Bergelson
Article copyright: © Copyright 2004 American Mathematical Society

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