Oppenheim conjecture for pairs consisting of a linear form and a quadratic form
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- by Alexander Gorodnik PDF
- 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.References
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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
- Email: gorodnik@math.ohio-state.edu, gorodnik@umich.edu
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 4447-4463
- MSC (2000): Primary 11J13, 11H55, 37A17
- DOI: https://doi.org/10.1090/S0002-9947-04-03473-7
- MathSciNet review: 2067128