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On a quantitative version of the Oppenheim conjecture
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## On a quantitative version of the Oppenheim conjecture

### Alex Eskin, Gregory Margulis, and Shahar Mozes

**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.

*Copyright American Mathematical Society 1996*

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#### Article Info

- ERA Amer. Math. Soc.
**01** (1995), pp. 124-130
- Publisher Identifier: S 1079-6762(95)03006-8
- 1991
*Mathematics Subject Classification*. Primary 11J25, 22E40.
- Received by the editors December 6, 1995
- Comments

**Alex Eskin**

Department of Mathematics, University of Chicago, Chicago, IL 60637,
USA

*E-mail address:* `eskin@math.uchicago.edu`

Research of the first author partially supported by an NSF
postdoctoral fellowship and by BSF grant 94-00060/1

**Gregory Margulis**

Department of Mathematics, Yale University, New Haven, CT 06520,
USA

*E-mail address:* `margulis@math.yale.edu`

Research of the second author partially supported
by NSF grants DMS-9204270 and DMS-9424613

**Shahar Mozes**

Institute of Mathematics, Hebrew University, Jerusalem 91904,
ISRAEL

*E-mail address:* `mozes@math.huji.ac.il`

Research of the third author partially supported by
the Israel Science foundation and by BSF grant 94-00060/1

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