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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