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

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

Published electronically:
January 13, 2004

MathSciNet review:
2067128

Full-text PDF

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]**A. Borel,*Values of indefinite quadratic forms at integer point and flows on spaces of lattices*, Bull. Amer. Math. Soc. 32 (1995), 184-204. MR**96d:22012****[Da00]**S. G. Dani,*On Values of linear and quadratic forms at integer points*, Number theory, pp. 107-119, Trends Math., Birkhäuser, Basel, 2000. MR**2001f:11108****[DM90]**S. G. Dani, G. A. Margulis,*Orbit closures of generic unipotent flows on homogeneous spaces of*, Math. Ann. 286 (1990), 101-128. MR**91k:22026****[Ma89]**G. Margulis,*Discrete subgroups and ergodic theory*, Number theory, trace formulas and discrete groups (Oslo, 1987), 377-398, Academic Press, Boston, MA, 1989. MR**90k:22013a****[Ma97]**G. A. Margulis,*Oppenheim Conjecture*, Fields Medalists' lectures, pp. 272-327, World Sci. Publishing, River Edge, NJ, 1997. MR**99e:11046****[OV]**A. Onishchik, E. Vinberg,*Lie groups and Lie algebras*, Encyclopedia Math. Sci. Vol. 41, Springer-Verlag, New York -- Berlin, 1994. MR**96d:22001****[Ra91]**M. Ratner,*Raghunathan's topological conjecture and distributions of unipotent flows*, Duke Math. J. 63 (1991), 235-280. MR**93f:22012****[Sh91]**N. A. Shah,*Uniformly distributed orbits of certain flows on homogeneous spaces*, Math. Ann. 289, 315-334, 1991. MR**93d:22010**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11J13,
11H55,
37A17

Retrieve articles in all journals with MSC (2000): 11J13, 11H55, 37A17

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