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10.3934/Dcds.2020359 DYNAMICAL SYSTEMS Volume 41, Number 5, May 2021 Pp DISCRETE AND CONTINUOUS doi:10.3934/dcds.2020359 DYNAMICAL SYSTEMS Volume 41, Number 5, May 2021 pp. 2205{2225 QUANTITATIVE OPPENHEIM CONJECTURE FOR S-ARITHMETIC QUADRATIC FORMS OF RANK 3 AND 4 Jiyoung Han Research Institute of Mathematics, Seoul National University GwanAkRo 1, Gwanak-Gu Seoul, 08826, South Korea (Communicated by Zhiren Wang) Abstract. The celebrated result of Eskin, Margulis and Mozes [8] and Dani and Margulis [7] on quantitative Oppenheim conjecture says that for irrational quadratic forms q of rank at least 5, the number of integral vectors v such that q(v) is in a given bounded interval is asymptotically equal to the volume of the set of real vectors v such that q(v) is in the same interval. In rank 3 or 4, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that two asymptotic limits coincide for almost all quadratic forms([8, Theorem 2.4]). In this paper, we extend this result to the S-arithmetic version. 1. Introduction. History. The Oppenheim conjecture, proved by Margulis [17], says that the image set q(Zn) of integral vectors of an isotropic irrational quadratic form q of rank at least 3 is dense in the real line (See [19] for the original statement of Oppenheim conjecture). Let (a; b) be a bounded interval and let Ω ⊂ Rn be a convex set containing the n origin. For any T 2 R>0, define T Ω = fv 2 R : v=T 2 Ωg. Dani and Margulis [7] and Eskin, Margulis and Mozes [8] established a quantitative version of Oppenheim conjecture: for a quadratic form q, let V(a;b);Ω(T ) be the volume of the region −1 −1 T Ω \ q (a; b) and N(a;b);Ω(T ) be the number of integral vectors in T Ω \ q (a; b). They found that if an irrational isotropic quadratic form q is of rank greater than or equal to 4 and is not a split form, then V(a;b);Ω(T ) approximates N(a;b);Ω(T ) as T goes to infinity. Moreover, in this case, the image set q(Zn) is equidistributed in the real line. Their works heavily rely on the dynamical properties of orbits of a certain se- misimple Lie group, generated by its unipotent one-parameter subgroups, on the associated homogeneous space. 2020 Mathematics Subject Classification. Primary: 22F30, 22E45, 20F65; Secondary: 20E08, 37P55. Key words and phrases. Quantitative Oppenheim conjecture, S-arithmetic space, action of quadratic form-preserving groups, low-dimensional p-adic symmetric space, Euclidean building. This paper is supported by the Samsung Science and Technology Foundation under project No. SSTF-BA1601-03 and the National Research Foundation of Korea(NRF) grant funded by the Korea government under project No. 0409-20200150. 2205 2206 JIYOUNG HAN The S-arithmetic space is one of the optimal candidates for a generalization of n n their work, since it has a lattice ZS which is similar to the integral lattice Z in Rn. Borel and Prasad [5] generalized Margulis' theorem to the S-arithmetic version and the author, Lim and Mallahi-Karai [13] proved the quantitative version of S- arithmetic Oppenheim conjecture. See also [29] and [15] for flows on S-arithmetic symmetric spaces. Ratner [21] generalized her measure rigidity of unipotent sub- groups in the real case to the Cartesian product of real and p-adic spaces. Set-up and notation. Consider a finite set Sf = fp1; : : : ; psg of odd primes and let S = f1g [ Sf . We refer to an element of S as a place. For each place p 2 S, denote by Qp the completion field of Q with respect to the p-adic norm. If p = 1, Q the norm k · k1 is the usual Euclidean norm and Q1 = R. Define QS = p2S Qp, the Cartesian product of Qp's (p 2 S). In this paper, we will use the product notations only for the Cartesian product. Since each Qp has the normed topology, let us assign the product topology of these normed topologies to QS. Let ∆ : Q ,! QS be the embedding given as ∆(z) = (z; : : : ; z) 2 QS, z 2 Q. Define n1 ns ZS = fmp1 ··· ps : m; n1; : : : ; ns 2 Zg ⊂ Q: It is well-known that the quotient =∆( ) ' = × Q is compact, where QS ZS R Z p2Sf Zp Zp = fz 2 Qp : jzjp ≤ 1g, so that ∆(ZS) is a uniform lattice subgroup of QS. For simplicity, we will use the notations Q and ZS instead of ∆(Q) and ∆(ZS), respectively. n Let us define an S-lattice Λ in QS as a free ZS-module of rank n. An S-quadratic form qS is an S-tuple of quadratic forms qp defined over Qp (p 2 S). We only consider the cases when all of qp have the same rank, and in this case, we define the rank of qS as the rank of qp. We call qS nondegenerate (isotropic, respectively) if every qp in qS is nondegenerate (isotropic, respectively). We say that qS is rational if there is a single rational quadratic form q such that qS = (λpq)p2S, where λp 2 Qp − f0g and qS is irrational if it is not rational. n Q n n One can regard a Q-module QS as the product p2S Qp of vector spaces Qp n (p 2 S). Let us denote an element of QS by v = (vp)p2S, where vp is a vector in n Qp for each p 2 S. We refer to the p-adic norm kvkp as the p-adic norm kvpkp of a n vector vp 2 Qp . Define the map σ : S ! f1; −1g by −1 if p < 1; σ(p) = 1 if p = 1: σ σ n We will use the notation v=kvkp := (vp=kvpkp )p2S for an element of QS consisting of unit vectors in each place p 2 S. n For p 2 S, we fix a star-shaped convex set Ωp on Qp centered at the origin defined as n σ Ωp = v 2 Qp : kvkp < ρp(v=kvkp ) ; n Q where ρp is a positive function on the set of unit vectors in Qp and let Ω = p2S Ωp. If p < 1, we further assume that ρp is (Zp − pZp)-invariant: for any u 2 Zp − pZp n n and for any unit vector vp 2 Zp − pZp , ρp(uvp) = ρ(vp): Let Ip ⊂ Qp be a bounded convex set of the form (a; b), where a; b 2 R, if p = 1, b and a+p Zp, where a 2 Qp and b 2 Z, if p < 1. We call Ip a p-adic interval. Define S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2207 an S-interval I = Q I . Denote by T = (T ) 2 × Q (pZ [ f0g), S p2S p p p2S R≥0 p2Sf m the S-tuple of radius parameters Tp, p 2 S, where pZ = fp : m 2 Zg. Consider Q TΩ = p2S TpΩp the dilation of Ω by T. Main statement. We are interested in the number N(T) = N(qS; Ω; IS)(T) of n elements v in ZS \TΩ such that qS(v) 2 IS. Let µ be the product of Haar measures µp on Qp, p 2 S. We assume that µ1 is the usual Lebesgue measure on R and µp(Zp) = 1 when p < 1. Let V(T) = V(qS; Ω; IS)(T) be the volume of the set n n fv 2 TΩ: qS(v) 2 ISg with respect to µ on QS. Recall that a quadratic form of rank 4 is split if it is equivalent to x1x4 − x2x3. In [13], when an isotropic irrational quadratic form qS is of rank n ≥ 4 and does not contain a split form, then as T ! 1, N(T) approximates V(T). Here, we say that T ! 1 if Tp ! 1 for all p 2 S. Moreover, it is possible to estimate V(T) in terms of I and T ([13, Proposition 1.2]). As a result, there is a constant λ > 0 S qS ;Ω such that as T goes to infinity, N(T) ∼ λ µ(I )jTjn−2; qS ;Ω S Q where jTj = p2S Tp ([13, Theorem 1.5]). Although [13] provides a precise condition of quadratic forms for which the quan- titative Oppenheim conjecture holds, for the case of rank 4, the condition that qS does not contain any split form, forces us to exclude substantial numbers of compo- s s s nents (3 × 7 − 2 × 6 among 3 × 7 , where s = #Sf ) in the space of nondegenerate n isotropic quadratic forms in QS. Note that each of the components can be identified 0 0 with the quotient space SO(qS)nSLn(QS), for some qS. In fact, there are quadratic forms for which N(T) fails to approximate V(T), when qS is of rank 3, or qS is of rank 4 and contains a split form. For instance, there is an irrational quadratic form 1−" qS of rank 3 and a sequence (Tj) for a given " > 0 so that N(Tj) ≥ jTjj(log jTjj) (See[8, Theorem 2.2] and [13, Lemma 9.2]). Our main theorem shows that quadratic forms for which N(T) approximates V(T) are generic in those excluded components, hence in the whole space of non- degenerate isotropic quadratic forms. In the case of rank 3, Theorem 1.1 is entirely a new result. Here, the term generic is with respect to the following measure: since one can 0 identify each component of the space of quadratic forms with SO(qS) n SLn(QS), 0 where qS = (qp)p2S is taken to be associated with the discriminant of each qp, one can assign an SLn(QS)-invariant measure on this space.
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