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 ∈ R>0, deﬁne T Ω = {v ∈ R : v/T ∈ Ω}. 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 semisimple 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 subgroups in the real case to the Cartesian product of real and p-adic spaces.

Set-up and notation. Consider a finite set Sf = {p1, . . . , ps} of odd primes and let S = {∞} ∪ Sf . We refer to an element of S as a place. For each place p ∈ S, denote by Qp the completion field of Q with respect to the p-adic norm. If p = ∞, Q the norm k · k∞ is the usual Euclidean norm and Q∞ = R. Define QS = p∈S Qp, the Cartesian product of Qp’s (p ∈ 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) ∈ QS, z ∈ Q. Define

n1 ns ZS = {mp1 ··· ps : m, n1, . . . , ns ∈ Z} ⊂ Q. It is well-known that the quotient /∆( ) ' / × Q is compact, where QS ZS R Z p∈Sf Zp Zp = {z ∈ Qp : |z|p ≤ 1}, 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 ∈ 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)p∈S, where λp ∈ Qp − {0} and qS is irrational if it is not rational. n Q n n One can regard a Q-module QS as the product p∈S Qp of vector spaces Qp n (p ∈ S). Let us denote an element of QS by v = (vp)p∈S, where vp is a vector in n Qp for each p ∈ S. We refer to the p-adic norm kvkp as the p-adic norm kvpkp of a n vector vp ∈ Qp . Define the map σ : S → {1, −1} by −1 if p < ∞, σ(p) = 1 if p = ∞. σ σ n We will use the notation v/kvkp := (vp/kvpkp )p∈S for an element of QS consisting of unit vectors in each place p ∈ S. n For p ∈ S, we fix a star-shaped convex set Ωp on Qp centered at the origin defined as n σ Ωp = v ∈ Qp : kvkp < ρp(v/kvkp ) , n Q where ρp is a positive function on the set of unit vectors in Qp and let Ω = p∈S Ωp. If p < ∞, we further assume that ρp is (Zp − pZp)-invariant: for any u ∈ Zp − pZp n n and for any unit vector vp ∈ Zp − pZp ,

ρp(uvp) = ρ(vp).

Let Ip ⊂ Qp be a bounded convex set of the form (a, b), where a, b ∈ R, if p = ∞, b and a+p Zp, where a ∈ Qp and b ∈ Z, if p < ∞. 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 ) ∈ × Q (pZ ∪ {0}), S p∈S p p p∈S R≥0 p∈Sf m the S-tuple of radius parameters Tp, p ∈ S, where pZ = {p : m ∈ Z}. Consider Q TΩ = p∈S 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) ∈ IS. Let µ be the product of Haar measures µp on Qp, p ∈ S. We assume that µ∞ is the usual Lebesgue measure on R and µp(Zp) = 1 when p < ∞. Let V(T) = V(qS, Ω, IS)(T) be the volume of the set n n {v ∈ TΩ: qS(v) ∈ IS} 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 → ∞, N(T) approximates V(T). Here, we say that T → ∞ if Tp → ∞ for all p ∈ 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 )|T|n−2, qS ,Ω S Q where |T| = p∈S Tp ([13, Theorem 1.5]). Although [13] provides a precise condition of quadratic forms for which the quantitative 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)\SLn(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−ε qS of rank 3 and a sequence (Tj) for a given ε > 0 so that N(Tj) ≥ |Tj|(log |Tj|) (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 nondegenerate 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) \ SLn(QS), 0 where qS = (qp)p∈S is taken to be associated with the discriminant of each qp, one can assign an SLn(QS)-invariant measure on this space.

Theorem 1.1. For almost all isotropic quadratic forms qS of rank 3 or 4, as T → ∞, N(q , Ω, I )(T) ∼ λ µ(I )|T|n−2, S S qS ,Ω S where n = rank(q ) and λ is a constant depending on the quadratic form q S qS ,Ω S and the convex set Ω described above. For the case of real quadratic forms of signature (2, 2), Eskin, Margulis and Mozes provided an example of a quadratic form such that Na,b,Ω(T ) does not approximate to Va,b,Ω(T ) and found a necessary condition of quadratic forms (called EWAS) that two quantities asymptotically coincide [9]. On the generic quadratic forms and their Oppenheim conjecture-type problems, there is a work of Bourgain [6] for real diagonal quadratic forms using the analytic number-theoretical method. Ghosh and Kelmer [11] showed another version of 2208 JIYOUNG HAN the quantitative version of Oppenheim problem for real generic ternary quadratic forms and Ghosh, Gorodnik and Nevo [10] extended this result for more general setting, such as for generic characteristic polynomial maps. Recently, there have been eﬀective results about the bound of the error terms N(a,b),Ω(T ) − V(a,b),Ω(T ) for almost all real quadratic forms of rank at least 3 by Athreya and Margulis [2], for almost all real homogeneous forms of even degree and of rank at least 3 by Kelmer and Yu [14], and for almost all systems of a homogeneous polynomial of even degree and a linear map with the certain condition by Bandi, Ghosh and the author [3]. For a pair of a quadratic form q and a linear form L of Rn, Gorodnik [12] computed the density of {(q(v),L(v)) : v ∈ P(Zn)} ⊆ R2 under the certain assumption and Lazar [16] generalized his result for the S-arithmetic setting. Sargent computed the density of M({v ∈ Zn : q(v) = a}) ⊆ R, where q is a rational quadratic form and M is a linear map [23]. In [24], he showed the quantitative version of his result.

Strategy and organization. The proof of the main theorem is provided in Section 4, by showing that V(T) approximates N(T) almost surely, using an equidistribution theorem (Proposition2) in one hand and the fact that V(T) is asymptotically n−2 proportional to the measure of IS and |T| on the other hand (Proposition4). Note that functions we consider in Section 4 are unbounded and non-compactly supported so that the general type of equidistribution theorems are not applicable.

In Section 3, we will introduce an αS-function, deﬁned on SLn(QS)/SLn(ZS), to modify an equidistribution theorem for those functions. The function αS gives an upper bound to functions we want to consider (so-called Schmidt’s lemma), hence enables us to ignore the cusp parts of such functions, in almost all cases. We will prove Proposition2 by using Howe-Moore theorem (Theorem 3.2) and Borel-Cantelli lemma. We also need to compute the measures of some given subsets of a maximal compact subgroup of SO(qS), which are associated with Cartan (also known as KAK) decomposition. For this, in Section 2, we ﬁrst review the well- known properties of symmetric spaces given by (products of) real hyperbolic spaces, and then explain how one can extend these properties to “the p-adic symmetric spaces”, which are Bruhat-Tits Euclidean buildings in our case.

2. Symmetric spaces. Let us denote by G an S-arithmetic group of the form Q G = p∈S Gp, where Gp is a semisimple Lie group deﬁned over Qp (p ∈ S). An element of G is g = (gp)p∈S = (g∞, g1, . . . , gs), where gi ∈ Gp . In most cases, G is Q i SLn(QS) := p∈S SLn(Qp), n ≥ 3. Consider the lattice subgroup Γ = SLn(ZS) of G and the symmetric space G/Γ, which can be embedded in the space of unimodular n S-lattices in QS. The Haar measure µG on G is the product of Haar measures µp on Gp. We will use the same notation µG for the normalized G-invariant measure on G/Γ. 0 When we refer to qS, which we will call a standard quadratic form, it is an S-tuple of quadratic forms consisting of: 2 q∞(x1, x2, x3) = 2x1x3 + x2, 0 2 if rank(qS) = 3; qp(x1, x2, x3) = 2x1x3 + a1x , 2 (1) 2 2 q∞(x1, x2, x3, x4) = 2x1x4 + a1x2 + a2x3, 0 2 2 if rank(qS) = 4, qp(x1, x2, x3, x4) = 2x1x4 + a1x2 + a2x3, where a1, a2 ∈ {±1} if p = ∞ and a1, a2 ∈ {±1, ±u0, ±p, ±pu0} if p < ∞. Here u0 is some ﬁxed square-free integer in Qp (See [26, Section 4.2]). S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2209

0 0 In SO(qS), we ﬁx a maximal compact subgroup K of SO(qS)

Y 0 Y 0 K = K∞ × Kp = SO(q∞) ∩ SO(n) × SO(qp) ∩ SLn(Zp) p∈Sf p∈Sf with the measure m = Q m , where m is the normalized Haar measure on K p∈Sf p p p (p ∈ S). We also ﬁx a diagonal subgroup    Y  A = a = a : t = (t , t , . . . , t ) ∈ × pZ × · · · × pZ , (2) t tpi ∞ 1 s R≥0 1 s  p∈S 

−t∞ t∞ tp −tp where at∞ = diag(e , 1,..., 1, e ) and atp = diag(p , 1,..., 1, p )(p ∈ Sf ). These groups K and A will be heavily used throughout the paper. We assume that T and t have the relation

t∞ tp T∞ = e and Tp = p , ∀p ∈ Sf ,

unless otherwise speciﬁed. Also we brieﬂy denote by Ti or ti instead of Tpi or tpi respectively. In this section, we take G∞ = SLn(R) and Gp = GLn(Qp), p < ∞, where n = 3 or 4. Corresponding maximal compact subgroups are Kˆ∞ = SO(n) and ˆ Kp = GLn(Zp), respectively.

3 0 2.1. The 3-dimensional hyperbolic space H . Let q∞ be a standard isotropic 0 quadratic form of rank n, n = 3, 4. The special orthogonal subgroup H∞ = SO(q∞) is one of the well-known linear groups SO(2, 1), SO(3, 1) and SO(2, 2). Let us examine the symmetric space K∞ \ H∞ and set a right invariant metric which is invariant by right multiplication. 3 3 Case i) H∞ = SO(3, 1). Set H to be the 3-dimensional hyperbolic space H = {z + ti : z = x + yj ∈ C and t ∈ R>0 }. The group SO(3, 1) is locally isomor- + 3 phic to PSL2(C), which is the group Isom (H ) of orientation-preserving isome- 3 3 tries of H . Here, the (P)SL2(C)-action on H is given by Mobius transformation, a b az + b .z = . Notice that entries of elements in SL ( ) are of the form c d cz + d n C x + yj. Since the stabilizer of the point i in PSL2(C) is the maximal compact subgroup PSU(2), we may identify the symmetric space K∞ \SO(3, 1), which is isomorphic to 3 PSU(2)\PSL2(C), with the hyperbolic space H . Let p : SO(3, 1) → K∞\SO(3, 1) ' 3 H be the projection given by p(g) = g.i. Deﬁne the metric d∞ of K∞ \ SO(3, 1) by

3 d∞(g1, g2) = d∞(K∞g1,K∞g2) := dH (p(g1), p(g2)).

Case ii) H∞ = SO(2, 1). Consider the identiﬁcation

2 K∞ \ SO(2, 1) ' PSO(2) \ PSL2(R) ' H = {x + ti : x ∈ R and t ∈ R>0 }.

Let us use the notation p for the projection SO(2, 1) → H2 as well. We deﬁne the metric d∞ of K∞ \ SO(2, 1) by

2 d∞(g1, g2) := dH (p(g1), p(g2)). 2210 JIYOUNG HAN

Case iii) H∞ = SO(2, 2). In the case of H∞ = SO(2, 2), consider the isomorphism 4 between real vector spaces R and M2(R) given by x y (x, y, z, w) 7→ . z w

x y The split form xw − yz of 4 corresponds to the determinant of in R z w M2(R). Moreover, there is the local isomorphism from SL2(R)×SL2(R) to SO(2, 2), which is induced by the action x y x y (g , g ). = g gt . (3) 1 2 z w 1 z w 2 Hence we deduce that

2 2 K∞ \ SO(2, 2) 'loc (SO(2) × SO(2)) \ (SL2(R) × SL2(R)) ' H1 × H2, 2 2 2 where H1 ' H ' H2. Note also that the action of at splits into the action of t/2 −t/2 (bt, bt), where bt = diag(e , e ). Put

d∞ = d := max{d 2 , d 2 }. SO(2,2) H1 H2

Lemma 2.1. Let H∞ be one of SO(2, 1), SO(3, 1) or SO(2, 2). Let K∞ be a maximal compact subgroup of H∞. Then there exists a constant C∞ > 0 such that for any t > 0 and for any r ∈ (0, 2t), −1 −2t+r m∞ k ∈ K∞ : d∞(atkat , 1) ≤ r < C∞e , (4) where m∞ is the normalized Haar measure on K∞. ∼ ∼ Proof. Assume that H∞ = SO(3, 1) = SL2(C) and K∞ = SU(2). For each t > 0, 3 K∞ acts transitively on the hyperbolic sphere S2t ⊂ H of radius 2t centered at i. Since d∞ is H∞-invariant, −1 m∞ k ∈ K∞ : d∞(atkat , 1) ≤ r = m∞ ({k ∈ K∞ : d∞(atk, at) ≤ r}) 2t 3 = LebS y ∈ S2t : dH (y, e i) ≤ r .

Here, LebS is the Lebesgue measure such that LebS(S2t) = 1. For the relation with 3 the p-adic case, p < ∞, let us mention that S2t is isomorphic to ∂H . Let x = e2ti and let θ be the angle between two geodesics from i to x and from i to y (Figure 1). By the hyperbolic law of cosines θ cosh(d(x, y)) = 1 + sinh2(2t) · (1 − cos θ) = 1 + 2 sinh2(2t) sin2( ), 2 and since d(x, y) ≤ r, we obtain that θ is bounded by e−2t+r/2. Hence

−1 2(−2t+r/2) m∞ k ∈ K∞ : d∞(atkat , 1) ≤ r ≤ C∞e for some constant C∞ > 0. ∼ The case of H∞ = SO(2, 1) = SL2(R) follows immediately from the compatibility between two embeddings SO(2, 1) ,→ SO(3, 1) and H2 ,→ H3 with the projection p (See also [8, Lemma 3.12]). ∼ If H∞ = SO(2, 2), using the local isometry given by (3), K∞ = SO(2) × SO(2) 2 2 acts transitively on the product St × St of two hyperbolic spheres in H × H . Thus S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2211 we have −1 m∞ k ∈ K∞ : d∞(atkat , 1) ≤ r

2 = mSO(2) ({k1 ∈ SO(2) : dH (btk1, bt) ≤ r})

2 × mSO(2) ({k2 ∈ SO(2) : dH (btk2, bt) ≤ r}) 2 < C e−t+r/2 = Ce−2t+r, where mSO(2) is the normalized Haar measure on SO(2).

H3 x y 2 H

Figure 1. The 3-dimensional hyperbolic space H3. The measure of the set in (4) is equal to the Lebesgue measure of the grey area on the top of the sphere.

2.2. A tree in the building Bn. For a prime p and n ≥ 2, let us brieﬂy recall the Euclidean building Bn, whose vertex set V(Bn) is isomorphic to GLn(Zp)\GLn(Qp). Recall the Cartan decomposition ([20, Theorem 3.14]) ˆ ˆ+ GLn(Qp) = GLn(Zp) · A · GLn(Zp) = GLn(Zp) · A · GLn(Zp), where ˆ m1 m2 mn A = {diag(p , p , . . . , p ): m1, m2, . . . , mn ∈ Z},

+ m1 m2 mn Aˆ = {diag(p , p , . . . , p ) : 0 ≤ m1 ≤ m2 ≤ · · · ≤ mn} ⊆ A.ˆ n Consider the space of free Zp-modules L of rank n in Qp on which GLn(Qp) acts by g.(Zpv1 ⊕ · · · ⊕ Zpvn) = Zp(v1g) ⊕ · · · ⊕ Zp(vng), n where g ∈ GLn(Qp) and v1,..., vn ∈ Qp forms an Zp-basis of L. Two rank-n free m Zp-modules L1 and L2 are said to be equivalent if L1 = p L2 for some m ∈ Z. The building Bn is an (n−1)-dimensional CW complex whose vertices are equiv- alence classes of free Zp-modules [L] of rank n. Vertices [L0], ...,[Lk] in Bn form a n k-simplex if there are representatives L0, ..., Lk in Qp such that pL0 ( L1 ( ··· ( Lk ( L0. 2212 JIYOUNG HAN

In particular, at each vertex [L] in Bn, the number of adjacent vertices with [L] n equals the number of proper nontrivial subspaces of the space (Zp/pZp) . Note that at most (n − 1) vertices form a simplex in Bn: there is an Zp-generating set {v1, v2,..., vn} of L such that

[pL] = [hpv1, pv2, . . . , pvni] ( [hv1, pv2, . . . , pvni] ( [hv1, v2, pv3, . . . , pvni] ( [hv1,..., vn−1, pvni ( [hv1,..., vni] = [L].

Vertices [L] in Bn is also denoted by n by n matrices whose rows are Zp-generators of L. By right multiplication of GLn(Zp), they are represented by upper triangle matrices whose diagonal entries are non-negative powers of p. In Bn, an apartment A is deﬁned as follows. The vertex set V(A) is ˆ V(A) = {[ag] ∈ GLn(Zp) \ GLn(Qp): a ∈ A} for some ﬁxed g ∈ GLn(Qp). A k-simplicial complex is in A if its vertices are all contained in A. Denote by A0 the apartment whose vertex set is

m1 mn {[diag(p , . . . , p )] : mi ∈ Z≥0} . n−1 n−1 Note that A is isometric to R and V(A0) is a lattice in R (See Figure 2), which is the reason that Bn is called a Euclidean building. There is the natural covering map p : Bn → A0 given by

 m1  p v12 ··· v1n m2  p ··· v2n  p :   7→ [diag(pm1 , pm2 , . . . , pmn )],  .. .   . .  pmn

mj where the matrix in the above map is a reduced type, i.e., mi ≥ 0 and gcd{p , vij} = 1, 1 ≤ i < j ≤ n. We give a distance d on V(Bn) by d([L1], [L2]) = `, where ` is the minimal number of edges connecting [L1] and [L2]. Let a : Z → V(A0) be any geodesic in the vertex set of A0. Then the inverse image −1 p ({a(m): m ∈ Z}) is a tree in Bn. If the above set {a(m)} is generated by one element, then the inverse image p−1({a(m)}) is a regular tree. More basic properties about buildings can be found in [1], [22] for Euclidean buildings, and see also [27] for Bruhat-Tits trees. Recall that a semisimple Lie group over a ﬁeld F is of F -rank m when its maximal abelian subgroup is locally isomorphic to F m.

Lemma 2.2. Let Hp be a Qp-rank one connected and simply connected semisimple algebraic subgroup of GLn(Qp). Denote the Cartan decomposition of Hp by Hp = ˆ Kp · A · Kp, where Kp = Hp ∩ GLn(Zp). Then the symmetric space Kp \ Hp is a tree.

Proof. One can embeds Kp \ Hp into GLn(Zp) \ GLn(Qp) by Kpg 7→ GLn(Zp)g, g ∈ Hp. Let us denote GLn(Zp)g by Kpg, g ∈ Hp, and {GLn(Zp)g : g ∈ Hp} by Kp \ Hp. ˆ Since Hp is of Qp-rank one, up to an appropriate conjugation, KpA is generated m1 mn by some element a = diag(p , . . . , p ) with m1 ≤ · · · ≤ mn and m1 6= 0. Hence ˆ we may assume that KpA = hai. Choose a geodesic {a(m): m ∈ Z} in V(A0) ˆ −1 containing KpA. Then Kp \ Hp is a subset of a tree p ({a(m): m ∈ Z}). S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2213

diag(1, p, p) diag(1, p, p2) ' diag(p−1, 1, p)

Id3 diag(1, 1, p)

2 Figure 2. Apartment A0 of B3. Kp \SO(2x1x3 −x2) is embedded in the inverse image of the blue line.

4 Let us show that for a nondegenerate isotropic quadratic form qp on Qp, if it is not a split form, then the special orthogonal group SO(qp) is of Qp-rank one, similar to the real case.

Lemma 2.3. Let qp be an isotropic non-split quadratic form of rank 4. If we ˆ ˆ denote SO(qp) = K · A · K, where K = SO(qp) ∩ SL4(Zp), then A is isomorphic to −t t the diagonal subgroup {diag(p , 1, 1, p ): t ∈ Z≥0}.

2 2 Proof. Without loss of generality, let qp(x, y, z, w) = 2xw + α1y + α2z , where α1, α2 ∈ {1, u0, p, u0p} for some ﬁxed 1 ≤ u0 ≤ p − 1, square-free over Qp. Since qp is not a split form, α2/α1 6= −1. Hence, up to a change of variables, we may further assume that −1 1. α2/α1 ∈ {p, pu0, pu0 } if −1 is square-free; −1 2. α2/α1 ∈ {p, pu0, pu0 , u0} otherwise. Since any semisimple element of a connected algebraic group is contained in a maximal abelian subgroup and two maximal abelian subgroups are conjugate to each other ([28, Theorem 6.3.5]), we may assume that

−t t ˆ diag(p , 1, 1, p ): t ∈ Z≥0 ⊆ A.

ˆ If the Qp-rank of A is larger than one, there is an element a = (aij)1≤i,j≤4 ∈ ˆ −t t A − SL4(Zp) such that a 6= diag(p , 1, 1, p ) for any t. From the commutativity, diag(p−t, 1, 1, pt) a = a diag(p−t, 1, 1, pt) for any t ∈ Z, it follows that      a11 0 0 0 a11 0 0 0 1 0 0 0  0 a22 a23 0   0 1 0 0  0 a22 a23 0  a = =   .  0 a32 a33 0   0 0 1 0  0 a32 a33 0  0 0 0 a44 0 0 0 a44 0 0 0 1 2214 JIYOUNG HAN

4 Since a ∈ SO(qp), it satisﬁes that for any (x, y, z, w) ∈ Qp,

2 2 2xw + α1y + α2z 2 2 = 2(a11x)(a44w) + α1(a22y + a23z) + α2(a32y + a33z) 2 2 2 = 2 (a11a44) xw + α1a22 + α2a32 y + 2 (α1a22a23 + α2a32a33) yz 2 2 2 + α1a23 + α2a33 z .

Since a11, a44 have no restriction except a11a44 = 1, by multiplying appropriate −t t −1 (p , 1, 1, p ) and (u, 1, 1, u ) to a, where t ∈ Z and u ∈ Zp − pZp, we may assume 4 that a11 = a44 = 1. Hence we need to ﬁnd (a22, a23, a32, a33) ∈/ Zp such that (a) 2 2 2 2 a22 + (α2/α1)a32 = 1, (b) a22a23 + (α2/α1)a32a33 = 0, (c) (α1/α2)a23 + a33 = 1. −1 −1 If α2/α1 = p, pu0, pu0 , let us denote α2/α1 = pu, where u ∈ {1, u0, u0 }. 2 2 2 P i Suppose that (a22, a32) ∈/ Zp and let a22 + pua32 = i≥m cip , where m ∈ Z is the ﬁrst term such that cm 6= 0. Denote by ν : Qp → Z the p-adic valuation. Then m/22 1. If |a22|p ≥ |a32|p, then m is even and ν(a22) < 0. Then cm = a22p m mod p and m = 2ν(a22) < 0 which contradicts to (a). (m−1)/22 2. If |a22|p < |a32|p, then m is odd and ν(a32) < 0. Then cm = u a32p m mod p and m = 2ν(a32) + 1 < 0 which is also impossible according to (a). 2 2 Hence (a22, a32) ∈ Zp. Similarly, (a23, a33) ∈ Zp. Now, assume α2/α1 = u0. In this case, −1 is a square. If there exists (a22, a23, 2 a32, a33) satisfying (a) through (c), then u0 = −(a22/a32) which contradict to the fact that u0 is square-free.

If qp is an isotropic quadratic form of rank 3, SO(qp) is isomorphic to SL2(Qp). Then the symmetric space Kp \ SO(qp) is isomorphic to a subtree of the (p + 1)- regular tree Tp. By Lemma 2.2 and Lemma 2.3, if qp is an isotropic non-split quadratic form of rank 4, then Kp \SO(qp) is a subtree of the building Bn. Lastly, if qp is a split form of rank 4, then SO(qp) is locally isomorphic to SL2(Qp) × SL2(Qp) as in (3). Then Kp \ SO(qp) is properly embedded to the product Tp × Tp of two (p + 1)-regular trees.

Corollary 1. Let p < ∞. Consider the p-adic element qp of a standard quadratic form of rank n = 3 or 4, given as in (1). Let Kp = SO(qp) ∩ SLn(Zp). For any r ∈ (0, 2t), p m k ∈ K : d (a ka−1, 1) ≤ r ≤ p−(2t−r), (5) p p p t t p + 1 where mp is the normalized Haar measure on Kp.

Proof. Suppose that qp is an isotropic quadratic form of rank 3 or of rank 4 and non-split. By Lemma 2.2 and Lemma 2.3, Kp \SO(qp) is isomorphic to a tree. Since dp is right Kp-invariant, the left hand side of (5) is

#{Kpatk : dtree(Kpatk, Kpat) ≤ r} mp ({k ∈ Kp : dp(atk, at) ≤ r}) = , #{Kpatk : k ∈ Kp} where #A is the cardinality of a (ﬁnite) set A. For any k ∈ Kp, consider the geodesic segment [Kp,Kpatk] in Tp. If dp(atk, at) ≤ r, since Kp \SO(qp) is a tree, [Kp,Kpatk] and [Kp,Kpat] have the common segment S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2215 of length at least 2t − dr/2e. Therefore

#{Kpatk : Kpat−dr/4ek = Kpat−dr/4e} mp ({k ∈ Kp : dp(atk, at) ≤ r}) = #{Kpatk : k ∈ Kp} 1 = , #{Kpat−dr/4ek : k ∈ Kp} where dre is the largest integer less than or equal to r. 2 Since SO(2xz − y )(Qp) is embedded in SO(qp), for any t ∈ N, there are at least 2 (t−1)−dr/4e (p + 1)p(p ) vertices in the sphere of radius 2t − dr/2e in Kp \ SO(qp). 2t −2t In the split case, the action of a2t = diag(p , 1, 1, p ) on Kp \ SO(qp) is con- ∼ verted to the action of (bt, bt) on SL2(Zp) \ SL2(Qp) × SL2(Zp) \ SL2(Qp) = Tp × Tp, t −t where bt = diag(p , p ). Then similar to the case of SO(2, 2)(R), we obtain the inequality (5).

3. Equidistribution property for generic points. For x ∈ G/Γ, let Λx be an n n n S-lattice in QS associated with x, i.e., if x = gΓ, Λx = gZS. We call L ⊆ QS a Λx-rational submodule if L is generated by elements in Λx. If L is a Λx-rational submodule,Λx ∩ L is an S-lattice in L. Let d(L) = dΛx (L) be a covolume of Λx ∩ L in L. If {v1,..., vj} is an ZS-basis of a Λx-rational submodule L of rank j, then d(L) is given by Y d(L) = kv1 ∧ · · · ∧ vjkp. p∈S Vj n Here for each p ∈ S, the p-adic norm k·kp of Qp is canonically extended to (Qp ), which we also denote by k · kp. Deﬁne a function αS : G/Γ → R>0 by 1 α (x) = sup : L is Λ -rational submodule . S d(L) x The following lemma originates from [25], stated for lattices in Rn. The statement and the proof of the S-arithmetic version can be found in [13].

n Lemma 3.1. For a bounded function f : QS → R of compact support, the Siegel transform f˜ of f is deﬁned by X f˜(x) = f(v), x ∈ G/Γ.

v∈Λx ˜ Then there is a constant Cf > 0 such that f(x) < Cf αS(Λx) for any x ∈ G/Γ. By Lemma 3.1 and Proposition1 below, one can easily deduce that a Siegel ˜ η transform f is in L (SLn(QS)/SLn(ZS)), 1 ≤ η < n.

Proposition 1. [13, Lemma 3.9] The function αS on SLn(QS)/SLn(ZS) belongs η to L (SLn(QS)/SLn(ZS)) for any real number 1 ≤ η < n, i.e., Z η |αS| dµG < ∞. SLn(QS )/SLn(ZS )

Here µG is the normalized SLn(QS)-invariant measure on SLn(QS)/SLn(ZS). In this section, we prove the equidistribution theorem for functions bounded by

αS. We ﬁrst recall the Howe-Moore theorem for SLn(QS). The statement for more general settings can be found in [4] or [18]. 2216 JIYOUNG HAN

Theorem 3.2. Consider G = SLn(QS) and Γ = SLn(ZS), n ≥ 3. Take a maximal ˆ ˆ Q ˆ Q 2 subgroup K of G as K = Kp = SO(n) × SLn(Zp). Let L0(G/Γ) be the space p∈S p∈Sf 2 R 2 of L -integrable functions f such that G/Γ f dµG = 0. 2 There exist constants λp > 0, p ∈ S satisfying the following: if f1, f2 ∈ L0(G/Γ) satisfy either

(1) both f1 and f2 are smooth of compact support or ˆ (2) both f1 and f2 are left K-invariant, then there exists a constant Cf1,f2 > 0 so that Z −λ∞d(g∞,1) Y −λpd(gp,1) f1(gx)f2(x)dµG(x) ≤ Cf1,f2 e p , G/Γ p∈Sf where g = (gp)p∈S ∈ G.

Recall that at is a diagonal element of A deﬁned in (2). For two S-tuples t = s 0 0 Q Z 0 0 (tp)p∈Sf and t = (tp)p∈Sf in R≥0 × i=1 pi , we say that t t if tp ≥ tp for all p ∈ S, and t t0 if t t0 and t 6= t0.

Lemma 3.3. Let qS be a nondegenerate isotropic quadratic form of rank n = 3 or 4. Let K be a maximal compact subgroup of SO(qS) as in Section 2. Set s = #Sf . Then there exist t0 0 and a constant CS > 0 such that for t t0 and for any r > 0,      X −1  s X m  k ∈ K : dp(atkat , 1) ≤ r  < CS r exp r − tp, (6)  p∈S  p∈S where m is the normalized Haar measure on K. P Proof. Consider (k1, . . . , ks) ∈ Z≥0 such that i ki ≤ r. From Lemma 2.1 and Lemma 2.2, d (a k a−1, 1) ≤ r − Ps k and ∞ t∞ ∞ t∞ i=1 i m k = (kp) ∈ K : −1 dpi (a kpi a , 1) ≤ ki, 1 ≤ i ≤ s tpi tpi s (r−P k )−t Y pi k −2t < C e i i ∞ × p i pi . ∞ p + 1 i i=1 i s The number of such nonnegative (k1, . . . , ks) is bounded above by r . Since e < p Q for any odd prime p, the inequality (6) holds for CS := C∞ i pi/(pi + 1).

We say that a sequence (tj) is divergent if (tj) escapes any bounded subset of × Q pZ as j → ∞. Recall that in Theorem 1.1, we are interested in the R>0 p∈Sf asymptotic limit of the given quantity when every tp of t goes to inﬁnity. However, for the proof of the main theorem, we need to consider more general concept of divergence in the next proposition and corollary.

Proposition 2. Let qS and K be as in Lemma 3.3 and let φ be a continuous function on G/Γ = SLn(QS)/SLn(ZS). Assume that there are constants η ∈ (0, n/2) and Cφ,η > 0 for which η |φ(x)| < Cφ,ηα(x)S, ∀x ∈ G/Γ. Then for any nonnegative bounded function ν on K and a divergent sequence (tj), Z Z Z

lim φ(atj kx0)ν(k)dm(k) = φ dµG ν dm (7) j→∞ K G/Γ K S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2217 for almost all x0 ∈ SLn(QS)/SLn(ZS). Proof. Let ε > 0 be arbitrary. For any function f on G/Γ, deﬁne Z Atf(x) = f(atkx)ν(k)dm(k). K P For t 0, let us denote by s(t) = p∈S tp. We claim that one can ﬁnd t0 0 and constants C, λ > 0 such that for any t t0, Z Z 0 −λs(t) µ Et = x ∈ G/Γ: At0 φ(x) − φ ν > ε for some t t < Ce , G which will imply that   R ∞ limj→∞ K φ(atj kx0)ν(k)dm(k) \ µG x : R R > ε ≤ µG  Etj  = 0. − φ dµG ν dm G/Γ K j=1

Let A(r) = {x ∈ G/Γ: αS(x) > r}. Choose a smooth non-negative function gr : G/Γ → [0, 1] such that

1. gr is K-invariant, i.e., gr(kx) = gr(x) for all k ∈ K; 2. gr(x) = 1 if x is in A(r + 1) and gr(x) = 0 if x is outside of A(r).

Take φ1 = φgr and φ2 = φ(1 − gr). Then φ = φ1 + φ2 and φ2 is compactly supported. Choose θ > 0 such that 1 ≤ η + θ < n/2. Then for any x ∈ G/Γ, η −θ η+θ φ1(x) = φ(x)gr(x) ≤ CφαS(x)gr(x) = CφαS (x)αS (x)gr(x) −θ η+θ ≤ Cφr αS (x). −θ η+θ Put ψ = Cφr αS (x). By the above inequalities and Proposition1, 1 2 |φ1| ≤ ψ and ψ ∈ L (G/Γ) ∩ L (G/Γ). Let f ∈ L2(G/Γ) be either a smooth function of compact support or a left 0 R R K-invariant function. Later, we will put f = φ2 − G/Γ φ2 or ψ − G/Γ ψ. By Theorem 3.2, there are λf , Cf > 0 such that for any g ∈ G,   Z X f(gx)f(x)dµG(x) ≤ Cf exp −λf dp(gp, 1) , (8) G/Γ p∈S where g = (gp)p∈S. For instance, we can take λf = min{λp, p ∈ S}, where λp’s are in Proposition 3.2. Note that Z Z 2 2 kAtfk2 = f(atkx)ν(k)dm(k) dµG(x) G/Γ K Z Z Z = f(atk1x)f(atk2x)ν(k1)ν(k2)dm(k2)dm(k1)dµG(x) G/Γ K K Z Z Z −1 −1 = f(atk1k2 at x)f(x)ν(k1)ν(k2)dm(k2)dm(k1)dµG(x) G/Γ K K Z Z Z −1 = f(atkat x)f(x)dµG(x) ν(kk2)ν(k2)dm(k2)dm(k) K G/Γ K Z Z ! −1 = f(atkat x)f(x)dµG(x) (ν ∗ νˆ)(k)dm(k), K G/Γ 2218 JIYOUNG HAN whereν ˆ(k) := ν(k−1), ∀k ∈ K. By Lemma 3.3,     X −1  s m U := k ∈ K : dp(atkat , 1) ≤ s(t)  ≤ CS (s(t)) exp (−s(t)) .  p∈S 

Hence by Cauchy-Schwartz inequality, we have that

Z Z −1 f(atkat x)f(x)dµG(x)(ν ∗ νˆ)(k)dm(k) U G/Γ (9) 2 s ≤ max(ν ∗ νˆ)kfk2 CS (s(t)) exp (−s(t)) . It is deduced from (8) that

Z Z −1 f(atkat x)f(x)dµG(x)(ν ∗ νˆ)(k)dm(k) K−U G/Γ (10)

≤ max(ν ∗ νˆ)Cf exp (−λf s(t)) .

1 1 1 1 Hence by combining (9) and (10), there are t0 0 and a constant λ = λ (f, t0) > 1 0 such that for t t0, 2 1 kAtfk2 ≤ exp(−λ s(t)). Now let us show following inequalities: ( Z Z )! ε 0 0 µG x ∈ G/Γ: At φ1 − φ1 ν > for some t t G/Γ K 2 (11)

< C1 exp(−λψs(t)) and ( Z Z )! ε 0 0 µG x ∈ G/Γ: At φ2 − φ2 ν > for some t t G/Γ K 2 (12)

< C2 exp(−λφ2 s(t)) for some positive constants C1 and C2. It is obvious that (11) and (12) imply (7). Q i First we note that αS(gx)/αS(x) ≤ max{ p∈S k ∧ (g)kp : i = 1, . . . , n}. Let tτ = (τ, 0,..., 0). Then there is M > 0 such that for all x ∈ G/Γ, for all τ ∈ [0, 1], η+θ η+θ αS (atτ x) ≤ MαS (x). Hence we have

ψ(atτ x) ≤ Mψ(x), ∀τ ∈ [0, 1].

Choose r > 0 suﬃciently large so that kφ1k1 ≤ kψk1 ≤ ε/(8 max(ν ∗νˆ)M). Then Z Z ε x ∈ G/Γ: At+tτ φ1 − φ1 ν > for some τ ∈ [0, 1] 2 n ε o ⊆ x ∈ G/Γ: |A φ | > for some τ ∈ [0, 1] t+tτ 1 4 n ε o n ε o ⊆ x ∈ G/Γ: |A φ | > ⊆ x ∈ G/Γ: |A ψ| > t 1 4M t 4M Z Z ε ⊆ x ∈ G/Γ: Atφ1 − ψ ν > . 8M S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2219

R By taking f = ψ − G/Γ ψ in (10) and using the Chebyshev’s inequality, there are ψ ψ t0 0 and λψ > 0 such that for any t t0 , Z Z ε µ E(t, t + t1) = x ∈ G/Γ: At+tτ φ1 − φ1 ν > for some τ ∈ [0, 1] G 2 8M 2 ≤ exp(−λ s(t)). ε ψ

Hence using the geometric series argument, there is a constant C1 > 0 such that the inequality (11) holds. Z Z ε 0 µ x ∈ G/Γ: At0 φ1 − φ1 ν > for some t t G 2 X X 0 0 ≤ |E(t , t + t1)| ≤ C1 exp(−λψs(t)). 0 0 t∞ = t∞ + n, ∀tp ≥ tp, ∀n ∈ N ∪ {0} p ∈ Sf

For (12), note that since φ2 is compactly supported, φ2 is uniformly continuous. Hence there is δ > 0 such that ε |A φ (x) − A φ (x)| < t+tτ 2 t 2 4 for all t 0, x ∈ G/Γ and τ ∈ [0, δ]. Again by (10) and Chebyshev’s inequality, φ2 φ2 there are t0 0 and λφ2 > 0 such that for all t t0 , ( )! Z Z ε µ x ∈ G/Γ: A φ − φ ν > for some τ ∈ [0, δ] G t+tτ 2 2 G/Γ K 2 ( )! Z Z ε 42 ≤ µ x ∈ G/Γ: A φ − φ ν > ≤ exp(−λ s(t)). G t 2 2 φ2 G/Γ K 4 ε Then (12) follows from the similar geometric argument used above.

n From now on, a function fp on Qp , p ∈ S, is always assumed to be compactly supported. If p < ∞, we additionally assume that fp is (Zp − pZp)-invariant: −1 fp(ux1, x2, . . . , xn−1, u xn) = fp(x1, x2, . . . , xn), ∀u ∈ Zp − pZp. (13)

Let ν be the product of non-negative continuous functions νp on the unit sphere n in Qp , p ∈ S. For p ∈ Sf , we also assume that

νp(uv) = νp(v), ∀u ∈ Zp − pZp. (14) Q Q Deﬁne a function Jf for f = p∈S fp by Jf = p∈S Jfp , where 1 Z Jf∞ (r, ζ∞) = n−2 f∞(r, x2, . . . , xn−1, xn)dx2 ··· dxn−1; r n−2 RZ −r 1 −r Jfp (p , ζp) = fp(p , x2, . . . , xn−1, xn) dx2 ··· dxn−1 (p ∈ Sf ), pr(n−2) n−2 Qp 0 −r where xn is determined by the equation ζp = q (p , x2, . . . , xn−1, xn) (If p = ∞, replace p−r by r). By Lemma 3.6 in [8] and Lemma 4.1 in [13], for suﬃciently small 2220 JIYOUNG HAN

0 0 ε > 0, there are c(Kp) > 0 and tp > 0 for each p ∈ S, such that if tp > tp, Z t∞(n−2) −1 c(K∞)e f∞(at∞ k∞v)ν(k e1)dm∞(k∞) ∞ K∞ (15) v −t∞ 0 −Jf∞ (kvk∞e , q∞(v))ν σ < ε; kvk∞

Z c(K )ptp(n−2) f (a k v)ν (k−1e )dm (k ) p p tp p p p 1 p p Kp (16) v tp σ 0 − Jfp (p kvkp , qp(v))ν σ < ε, p ∈ Sf . kvkp Recall that σ = 1 for the infinite place and σ = −1 for the finite place. Define σ σ σ σ kgvk /T = (kgpvkp /Tp )p∈S. By Lemma 3.6 in [8] and Lemma 4.1 in [13], there is a constant c(K) > 0 such that for sufficiently small ε > 0 and sufficiently large t 0, σ kgvk gv Jf , q (gv) ν Tσ S kgvkσ Z (17) n−2 ˜ −1 −c(K)|T| f(atkgΓ)ν(k e1)dm(k) ≤ ε. K

Corollary 2. Let (Tj = (Tj,p)p∈S)j∈N be a divergent sequence. Deﬁne 0 S = {p ∈ S :(Tj,p)j∈N is bounded } ( S.

Let Ω, IS = (Ip)p∈S be as in Theorem 1.1. Then for almost all nondegenerate 0 isotropic quadratic form qS = (qp)p∈S, there is a constant C = C(S ) > 0 such that n Y n−2 # {v ∈ ZS ∩ TjΩ: qS(v) ∈ IS} < C (Tj,p) , p∈S−S0 where n is the rank of qS. Proof. Since we want to show the upper bound, for simplicity, we may assume that n Ω is the product of unit balls in Qp , p ∈ S. Let ε > 0 be given. For each p ∈ S, let Cp be a compact set of the space of nondegenerate isotropic quadratic forms over Qp of a given signature. Let C = Q p∈S Cp. Let g be an element of SL ( ) such that q (v) = q0 (g v) for all v ∈ n. qS n QS S S qS QS −1 Then for each p ∈ S, there is βp = βp(Cp) > 0 such that if qS ∈ C, βp ≤ kg vk /kvkσ ≤ β for all v ∈ n. qS p p p QS Choose bounded continuous functions fp, p ∈ S, of compact support such that 1 Jfp ≥ 1 + ε on [ , βp] × Ip. 2βp

If v satisﬁes that Tp/2 ≤ kvkp ≤ Tp and qp(v) ∈ Ip, then σ 0 Jfp (kgvkp/Tp , qp(gv)) ≥ 1 + ε for g = g with q ∈ C. By (15) and (16), there is T0 = (T 0) 0 such that for qS S p p∈S 0 each p ∈ S and for all Tp > Tp ,

Z kvkσ c(K )T n−2 f (a k v)dm (k ) − J p , ζ < ε. p p p tp p p p fp T σ Kp p S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2221

0 0 For p ∈ S , we may further assume that Tj,p ≤ Tp for all j. n Finally for each p ∈ S, choose a nonnegative bounded function gp on Qp of 0 0 compact support such that if kvkp ≤ Tp and tp ≤ logp(Tp ), Z

gp(atp kpv)dmp(kp) ≥ 1. Kp Q Q Take hS0 = p∈S0 gp × p∈S−S0 fp and let hfS0 be the Siegel transform of hS0 0 0 deﬁned in Lemma 3.1. Then for T = (Tp) such that Tp > Tp , ∀p ∈ S − S , we obtain that 0 0 n kvkp ≤ Tp , ∀p ∈ S # v ∈ ZS : 0 , qS(v) ∈ IS Tp/2 < kvkp ≤ Tp, ∀p ∈ S − S   X Y Z Y Z ≤ g (a k g v)dm (k ) × T n−2 f (a k g v)dm (k )  p tp p qS p p p p tp p qS p p  n 0 Kp 0 Kp v∈ZS p∈S p∈S−S   Z Y n−2 = T h 0 (a kg v)dm(k).  p  fS t qS p∈S−S0 K

By Proposition2, for almost all quadratic form qS, as t diverges, Z Z h 0 (a kg SL ( ))dm(k) → h 0 dµ < ∞. fS t qS n ZS fS G K G/Γ 0 Hence there is a constant C > 0 such that for all Tj, 0 0 n kvkp ≤ Tp , ∀p ∈ S 0 Y n−2 # v ∈ ZS : 0 , qS(v) ∈ IS < C (Tj,p) . Tj,p/2 < kvkp ≤ Tj,p, ∀p ∈ S − S p∈S−S0 Therefore we have    n−2 Y p Y # {v ∈ n ∩ T Ω: q (v) ∈ I } < C0 T . ZS j S S  p − 1  j,p p∈S−S0 p∈S−S0

4. The proof of the main theorem. The proof of Theorem 1.1 is similar to that of Theorem 1.5 in [13] except we use Proposition2 instead of Theorem 7.1 in [13]. Let C be any compact subset of the space of isotropic quadratic forms with equal signature and let ε > 0 be given. Let φ and ν be compactly supported functions defined on G/Γ and K, resepctively, as before. Theorem 7.1 in [13] says that there is t0 = t0(C) 0 such that except on a finite 0 union of orbits of SO(qS), x ∈ C satisfies the following: for all t t0,

Z Z Z

φ(atkx)ν(k)dm(k) − φdµG νdm < ε. K G/Γ K Proposition 3. Let q be an isotropic quadratic form of rank n = 3 or 4. Let Q Q S f = fp and ν = νp be as in (13) and (14). Assume further that f satisﬁes the following condition: there is a nonnegative continuous function f + of compact n + ◦ ◦ support on QS such that supp(f) ⊂ supp(f ) , where A is an interior of A and Z + sup ff (atkgΓ)dm(k) = M < ∞. (18) t 0 K 2222 JIYOUNG HAN

Then there exists t0 0 such that if t t0,

σ −(n−2) X kgvk gv |T| Jf σ , qS(gv) ν σ n T kgvk v∈ZS (19) Z ˜ −1 −c(K) f(atkg)ν(k e1)dm(k) ≤ . K Proof. We ﬁrst claim that there is a constant c > 0 such that kgvkσ gv # Π := v ∈ n : J , q (gv) ν 6= 0 < c|T|n−2. (20) ZS f Tσ S kgvkσ Since the interior of supp(f +) contains supp(f), there is ρ > 0 such that

Jf + > ρ on supp(Jf ).

Note that since Jf + is compactly supported, the set Π is finite. Take ε > 0 such that ε < ρ/2. Then by (17), for sufficiently large t, we have that X kgvkσ ρ × (#Π) ≤ J + , q (gv) f Tσ S v∈Π Z n−2 + ≤ c(K)|T| ff (atkgΓ)dm(k) + ε × (#Π). K By (18), ρ/2 × (#Π) ≤ c(K)M|T|n−2. This implies (20). Now (19) follows from applying (17) by putting ε/(c(K)ρM) instead of ε and n taking summation over all v ∈ ZS. Recall that a convex set Ω is defined using a non-negative continuous function Q n ρ = p∈S ρp, where ρp is a positive function on the unit sphere of Qp , p ∈ S. Define the shell Ωˆ of Ω by ˆ n σ σ Ω = v ∈ QS : ρp(v/kvkp )/2 < kvkp ≤ ρp(v/kvkp ), ∀p ∈ S . Note that when p < ∞, the inequality in the above definition is in fact the σ equality: kvkp = ρp(v/kvkp ). The following proposition was originally stated for Ω but the proof can be easily modified for Ω.ˆ

Proposition 4. [13, Proposition 1.2] There is a constant λ = λ(qS, Ω) > 0 such that as T → ∞, n n n ˆ o ˆ n−2 µ v ∈ QS ∩ TΩ: qS(v) ∈ IS ∼ λ(qS, Ω) · µ(IS) · |T| , n n where µ and µ are Haar measures of QS and QS, respectively, which are deﬁned in Section 1. Lemma 4.1. [13, Lemma 5.2] Let f and ν be as in Proposition3. Take σ σ hp(vp, ζp) = Jfp (kvpkp , ζp)νp(vp/kvpkp ), p ∈ S Q and set h(v, ζ) = p∈S hp(vp, ζp). Then we have Z −(n−2) v n lim |T| h σ , qS(v) dµ (v) T→∞ n T QS Z Z (21) ˜ Y −1 = c(K) f dµG νp(kp e1)dmp(kp). G/Γ p∈S Kp S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2223

Proof of Theorem 1.1. Since f in Proposition3 is compactly supported, there is f + + ◦ + such that supp(fp) ⊂ supp(fp ) . By Proposition2 with φ = f and ν ≡ 1, Z Z + + lim ff (atkgΓ)dm(k) = ff dµG t→∞ K G/Γ for almost all x = gΓ ∈ G/Γ. By Lemma 3.1 and Proposition1, the integral of ff+ over G/Γ is ﬁnite. Hence one can apply Proposition3 for almost all x ∈ G/Γ. Q We also remark that the set of functions of the form Jf ν, where f = fp, Q ν = νp with (13) and (14) respectively, is a generating set of n L = {F (v, ζ):(QS) × QS → R | F (uv, ζ) = F (v, ζ), ∀u ∈ Zp − pZp, ∀p ∈ Sf }. Hence Proposition3 holds for functions in L as well (See details in [13]). Deﬁne Z −(n−2) v n L(h) := lim |T| h σ , qS(v) dµ (v), h ∈ L. T→∞ n T QS The characteristic function I (gv, ζ) is contained in L. Let ε > 0 be given. TΩˆ×IS a b Take continuous functions h , h ∈ L, depending on T, ε and qS such that hb(gv, ζ) ≤ I (gv, ζ) ≤ ha(gv, ζ) and L(ha) − L(hb) < ε. (22) TΩˆ×IS a b From (19), (7) and (21), for h = h , h and any ε > 0, there is T0 0 such that if T T , 0

−(n−2) X gv |T| h σ , qS(gv) − L(h) < ε (23) n T v∈ZS for almost every gΓ ∈ G/Γ. By the deﬁnition of L(h) and rescaling T0 0 if necessary, we also obtain that

Z gv −(n−2) |T| h σ , qS(gv) − L(h) < ε. (24) n T QS Combining (23) and (24) with (22), if we regard h as the characteristic function of TΩˆ × IS, n o n o n ˆ n ˆ # v ∈ ZS ∩ TΩ: qS(v) ∈ IS − vol v ∈ QS ∩ TΩ: qS(v) ∈ IS < 4ε. Since n #({v ∈ ZS ∩ TΩ: qS(v) ∈ IS}) n o X n −n0 −n1 −ns ˆ = # v ∈ ZS ∩ (2 T∞, p1 T1, . . . , ps Ts)Ω: qS(v) ∈ IS , nj ∈ {0} ∪ N j ∈ {0, . . . , s} the lemma follows from Proposition4 and the classical argument of geometric series, if we obtain the following: as T → ∞, the summation over all S = (S∞,S1,...,Ss) = −n0 −n1 −ns 0 0 0 (2 T∞, p1 T1, . . . , ps Ts) with S T0 = (T∞,T1 ,...,Ts ) is X n n ˆ o n−2 # v ∈ ZS ∩ SΩ: qS(v) ∈ IS = o(|T| ) as T → ∞. S For this, by rescaling T0 if necessary, let us assume that Corollary2 holds for 0 −n0 −n1 −ns any S ( S. Denote the set {S = (2 T∞, p1 T1, . . . , ps Ts) ≺ T : S T0} by 0 ∪{ΨS0 : S ⊆ S}, where 0 0 0 0 ΨS0 := S = (S∞,S1,...,Ss): Sp ≤ Tp , ∀p ∈ S and Sp > Tp , ∀p ∈ S − S . 2224 JIYOUNG HAN

Then for S0 ( S, by Corollary2, there is a constant C(S0) > 0 such that X n n ˆ o n # v ∈ ZS ∩ SΩ: qS(v) ∈ IS < #({v ∈ ZS ∩ TS0 Ω: qS(v) ∈ IS}) S∈Ψ 0 S (25) 0 Y n−2 < C(S ) (Tp) , p∈S−S0 0 0 0 0 0 0 where TS0 = (Tp)p∈S. Here Tp = Tp for p ∈ S and Tp = Tp for p ∈ S − S . Hence Q n−2 0 the left hand side of (25) is o( p∈S−S0 Tp ). If S = S, since X n n ˆ o n # v ∈ ZS ∩ SΩ: qS(v) ∈ IS ≤ #({v ∈ ZS ∩ T0Ω: qS(v) ∈ IS}) < ∞, S∈ΨS it is obviously o(|T|n−2).

Acknowledgments. I would like to thank Seonhee Lim and Keivan Mallahi-Karai for suggesting this problem and providing valuable advices. This paper is supported by the Samsung Science and Technology Foundation under project No. SSTF- BA1601-03.

REFERENCES

[1] P. Abramenko and K. S. Brown, Buildings. Theory and Applications, Graduate Texts in Mathematics, 248. Springer, New York, 2008. [2] J. S. Athreya and G. A. Margulis, Values of random polynomials at integer points, J. Mod. Dyn., 12 (2018), 9–16. [3] P. Bandi, A. Ghosh and J. Han, A generic effective Oppenheim theorem for systems of forms, J. Number Thoery, 218 (2020), 311–333. [4] Y. Benoist, Five lectures on lattices in semisimple Lie groups, GéométriesàCourbure Négative ou Nulle, Groupes Discrets et Rigidités,Sémin.Congr., Soc. Math. France, Paris, 18 (2009), 117–176. [5] A. Borel and G. Prasad, Values of isotropic quadratic forms at S-integral points, Compos. Math., 83 (1992), 347–372. [6] J. Bourgain, A quantitative Oppenheim theorem for generic diagonal quadratic forms, Israel J. Math., 215 (2016), 503–512. [7] S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel’fand Seminar, Adv. Soviet Math., Part 1, Amer. Math. Soc., Providence, RI, 16 (1993), 91–137. [8] A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93–141. [9] A. Eskin, G. Margulis and S. Mozes, Quadratic forms of signature (2, 2) and eigenvalue spacings on rectangular 2-tori, Ann. of Math., 161 (2005), 679–725. [10] A. Ghosh, A. Gorodnik and A. Nevo, Optimal density for values of generic polynomial maps, preprint, arXiv:1801.01027. [11] A. Ghosh and D. Kelmer, A quantitative Oppenheim theorem for generic ternary quadratic forms, J. Mod. Dyn., 12 (2018), 1–8. [12] A. Gorodnik, Oppenheim conjecture for pairs consisting of a linear form and a quadratic form, Trans. Amer. Math. Soc., 356 (2004), 4447–4463. [13] J. Han, S. Lim and K. Mallahi-Karai, Asymptotic distribution of values of isotropic quadratic forms at S-integral points, J. Mod. Dyn., 11 (2017), 501–550. [14] D. Kelmer and S. Yu, Values of random polynomials in shrinking targets, preprint, arXiv:1812.04541. [15] D. Kleinbock and G. Tomanov, Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv., 82 (2007), 519–581. [16] Y. Lazar, Values of pairs involving one quadratic form and one linear form at S-integral points, J. Number Theory, 181 (2017), 200–217. [17] G. A. Margulis, Formes quadratriques indéfinieset flots unipotents sur les espaces homogénes, C. R. Acad. Sci. Paris. Sér. I Math., 304 (1987), 249–253. S-ARITHMETIC QUANTITATIVE OPPENHEIM FOR LOW RANK 2225

[18] H. Oh, Uniform pointwise bounds for matrix coeﬃcients, Duke Math. J., 113 (2002), 133–192. [19] A. Oppenheim, The Minima of Indeﬁnite Quaternary Quadratic Forms, Thesis (Ph.D.)–The University of Chicago, 1930. [20] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics, 139. Academic Press, Inc., Boston, MA, 1994. [21] M. Ratner, Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups, Duke Math. J., 77 (1995), 275–382. [22] G. Robertson, Euclidean Buildings, (lecture), “Arithmetic Geometry and Noncommutative Geometry”, Masterclass, Utrecht, 2010. [23] O. Sargent, Density of values of linear maps on quadratic surfaces, J. Number Theory, 143 (2014), 363–384. [24] O. Sargent, Equidistribution of values of linear forms on quadratic surfaces, Algebra Number Theory, 8 (2014), 895–932. [25] W. M. Schmidt, Approximation to algebraic numbers, Enseignement Math., 17 (1971), 187– 253. [26] J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973. [27] J.-P. Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. [28] T. A. Springer, Linear Algebraic Groups, Second edition, Progress in Mathematics, 9, Birkh user Boston, Inc., Boston, MA, 1998. [29] G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 26 (2000), 265–297. Received January 2020; revised August 2020. E-mail address: [email protected]