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 | References | Similar Articles | Additional Information

Abstract: Let be a nondegenerate quadratic form and a nonzero linear form of dimension . As a generalization of the Oppenheim conjecture, we prove that the set is dense in provided that and satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.

**[Bo95]**Armand Borel,*Values of indefinite quadratic forms at integral points and flows on spaces of lattices*, Bull. Amer. Math. Soc. (N.S.)**32**(1995), no. 2, 184–204. MR**1302785**, 10.1090/S0273-0979-1995-00587-2**[Da00]**S. G. Dani,*On values of linear and quadratic forms at integral points*, Number theory, Trends Math., Birkhäuser, Basel, 2000, pp. 107–119. MR**1764798****[DM90]**S. G. Dani and G. A. Margulis,*Orbit closures of generic unipotent flows on homogeneous spaces of 𝑆𝐿(3,𝑅)*, Math. Ann.**286**(1990), no. 1-3, 101–128. MR**1032925**, 10.1007/BF01453567**[Ma89]**G. A. Margulis,*Discrete subgroups and ergodic theory*, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 377–398. MR**993328****[Ma97]**G. A. Margulis,*Oppenheim conjecture*, Fields Medallists’ lectures, World Sci. Ser. 20th Century Math., vol. 5, World Sci. Publ., River Edge, NJ, 1997, pp. 272–327. MR**1622909**, 10.1142/9789812385215_0035**[OV]**È. B. Vinberg (ed.),*Lie groups and Lie algebras, III*, Encyclopaedia of Mathematical Sciences, vol. 41, Springer-Verlag, Berlin, 1994. Structure of Lie groups and Lie algebras; A translation of Current problems in mathematics. Fundamental directions. Vol. 41 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [ MR1056485 (91b:22001)]; Translation by V. Minachin [V. V. Minakhin]; Translation edited by A. L. Onishchik and È. B. Vinberg. MR**1349140****[Ra91]**Marina Ratner,*Raghunathan’s topological conjecture and distributions of unipotent flows*, Duke Math. J.**63**(1991), no. 1, 235–280. MR**1106945**, 10.1215/S0012-7094-91-06311-8**[Sh91]**Nimish A. Shah,*Uniformly distributed orbits of certain flows on homogeneous spaces*, Math. Ann.**289**(1991), no. 2, 315–334. MR**1092178**, 10.1007/BF01446574

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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

DOI:
https://doi.org/10.1090/S0002-9947-04-03473-7

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