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ISSN 1079-6762

On a quantitative version of the Oppenheim conjecture


Authors: Alex Eskin, Gregory Margulis and Shahar Mozes
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 124-130
MSC (1991): Primary 11J25, 22E40
MathSciNet review: 1369644
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Abstract: The Oppenheim conjecture, proved by Margulis in 1986, states that the set of values at integral points of an indefinite quadratic form in three or more variables is dense, provided the form is not proportional to a rational form. In this paper we study the distribution of values of such a form. We show that if the signature of the form is not $(2,1)$ or $(2,2)$, then the values are uniformly distributed on the real line, provided the form is not proportional to a rational form. In the cases where the signature is $(2,1)$ or $(2,2)$ we show that no such universal formula exists, and give asymptotic upper bounds which are in general best possible.


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Additional Information

Alex Eskin
Affiliation: Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
Email: eskin@math.uchicago.edu

Gregory Margulis
Affiliation: Department of Mathematics, Yale University, New Haven, CT, USA
Email: margulis@math.yale.edu

Shahar Mozes
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, ISRAEL
Email: mozes@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S1079-6762-95-03006-X
PII: S 1079-6762(95)03006-X
Received by editor(s): December 6, 1995
Additional Notes: Research of the first author partially supported by an NSF postdoctoral fellowship and by BSF grant 94-00060/1
Research of the second author partially supported by NSF grants DMS-9204270 and DMS-9424613
Research of the third author partially supported by the Israel Science foundation and by BSF grant 94-00060/1
Article copyright: © Copyright 1996 American Mathematical Society