tailieunhanh - Đề tài " Quadratic forms of signature (2, 2) and eigenvalue spacings on rectangular 2tori "

The Oppenheim conjecture, proved by Margulis [Mar1] (see also [Mar2]), asserts that for a nondegenerate indefinite irrational quadratic form Q in n ≥ 3 variables, the set Q(Zn ) is dense. In [EMM] (where we used the lower bounds established in [DM]) a quantitative version of the conjecture was established. Namely: Let ρ be a continuous positive function on the sphere {v ∈ Rn | v = 1}, and let Ω = {v ∈ Rn | v | Annals of Mathematics Quadratic forms of signature 2 2 and eigenvalue spacings on rectangular 2-tori By Alex Eskinn Gregory Margulis andShahar Mozes Annals of Mathematics 161 2005 679 725 Quadratic forms of signature 2 2 and eigenvalue spacings on rectangular 2-tori By Alex Eskin GREGORy Margulis and Shahar Mozes 1. Introduction The Oppenheim conjecture proved by Margulis Marl see also Mar2 asserts that for a nondegenerate indefinite irrational quadratic form Q in n 3 variables the set Q Zn is dense. In EMM where we used the lower bounds established in DM a quantitative version of the conjecture was established. Namely Let p be a continuous positive function on the sphere v E Rn I IIvII 1 and let Q v E Rn I vh p v v . We denote by TQ the dilate of Q by T. For an indefinite quadratic form Q in n variables let NQ a a b T denote the cardinality of the set x E Zn x E TQ and a Q x b . We recall from EMM that for any such Q there exists a constant Aq q such that for any interval a b as T TO 1 Vol x E Rn x E TQ and a Q x b ẦQ Q b - a Tn-2. Theorem EMM Th. . Let Q be an indefinite quadratic form of signature p q with p 3 and q 1. Suppose Q is not proportional to a rational form. Then for any interval a b as T TO 2 NQ Q a b T Q Q b - a Tn- 2 where n p q and Aq q is as in 1 . If the signature of Q is 2 1 or 2 2 then Theorem fails in fact there are irrational forms for which along a subsequence Tj NQ n a b Tj Tn-2 logTj 1- . Such forms may be obtained by consideration of irrational Research partially supported by BSF grant 94-00060 1 GIF grant 95 the Sloan Foundation and the Packard Foundation. Research partially supported by NSF grant DMS-9424613. Research partially supported by the Israel Science Foundation and by BSF grant 9400060 1. 680 ALEX ESKIN GREGORY MARGULIS AND SHAHAR MOZES forms which are very well approximated by split rational forms. It should be noted that the asymptotically exact lower bounds established by Dani and Margulis see DM .