Effective equidistribution of S-integral points on symmetric varieties

EFFECTIVE EQUIDISTRIB UTI ON OF S -INTEGRAL POIN T S ON SYMMETRIC V ARIETI E S YVES BENOIST AND HEE OH Abstract. Let K b e a global ﬁeld of characteris tic not 2. Let Z = H \ G b e a symmetric v ariety deﬁned ov e r K and S a ﬁnite set of places of K . W e obtain counting and e q uidistribution results for the S -integral p oints of Z . Our re s ults ar e eﬀective when K is a num b er ﬁeld. 1. Introductio n 1.1. General o verview. Consider a ﬁnite system of p olynomial equations with in tegral co eﬃcien ts. Its set of solutions deﬁnes an arit hmetic v ariet y Z ⊂ C d deﬁned ov er Z . F or a set S of primes including the inﬁnite prime ∞ , let Z S denote the ring of S -in tegers of Q , that is, the set of rational n um b ers whose denominators are pro ducts of primes in S . If S = { ∞} , Z S is simply the ring o f in tegers Z , and if S consists of all the primes, then Z S is the ﬁeld Q of ratio nal n umbers. F or an y subring R of C , w e denote b y Z R the set of p o in ts in Z with co ordinates in R . One o f the fundamen tal questions in n um b er theory is to understand the prop erties of sets Z Z S . In this pap er, we obtain eﬀectiv e coun ting and equidistribution results of the S -in tegr a l p oin ts, for S -ﬁnite, in the case when Z is a symmetric v a r iety . The coun ting question in this set-up has been completely solv ed for the in- te gr a l p oints via sev eral diﬀeren t metho ds. The ﬁrst solution is due to Duk e, Rudnic k and Sa r na k in 1993 [15] and their pro of uses the theory of automorphic forms. Almost at the same time, Es kin and McMullen ga v e t he second pro of utilizing mixing properties of semis imple real algebraic groups [17]. The t hird pro of, due to Eskin, Mozes a nd Shah [18], is based on the ergo dic theory of ﬂo ws on ho mogeneous spaces, more precisely , Ratner’s w or k on the unip oten t ﬂows . The approac h of [17] using mixing prop erties has sev eral adv an tages ov er the others in our viewpoint. First it do es not require the deep theory of automor phic forms, av oiding tec hnical diﬃculties in dealing with the Eisenstein series as in [15]. Secondly , although this w as nev er addressed in [17], in principle it also giv es a ra te of con vergenc e whic h the ergo dic metho d of [18 ] do es not giv e. Thirdly the metho d can b e extended to other global ﬁelds o f p ositiv e c haracteristic, whic h is again hard to b e ach iev ed via the ergo dic metho d. the second author is pa rtially supp or ted by NSF gr ant 062932 2. 1 2 YVES BENOIST AND HEE OH F or these reasons, w e dev elop the approach of Eskin and McMullen [17] in this pap er in order t o obtain eﬀe ctive results for the general S - in tegral p oin ts on symmetric v arieties. W e use the mixing prop erties of S - algebraic semisimple groups, with a rate of conv ergence. Implemen ting this in the coun ting problem, a crucial tec hnical ingredien t is to verify certain geometric prop ert y , whic h w a s na med the w av e- fron t pro p ert y b y [17], fo r an S -algebraic symmetric v ariet y . W e pro v e this using the p olar decomp ositions for non-arc himedean symmetric spaces obta ined in [4 ] sp eciﬁcally for this purp ose. W e emphasize that the wa v e front prop erty is pre- cisely the r eason that our pro of s w ork in the setting of a n S -algebraic symmetric v ariet y for S ﬁnite. This prop ert y do es not hold f or a general homo g eneous v ari- et y ev en o ve r the reals. In obt a ining eﬀectiv e coun ting results fo r the S - in tegral p oin ts of b ounded heigh t, w e also use the w orks of Denef on p - adic lo cal zeta functions ([12], [13]) and of Jeanquartier on ﬁb er in tegrat io ns [30]. W e remark t ha t t he approa c h fo r coun ting via mixing w as initiated in 19 70 by Margulis in his dissertation on Anoso v dynamical systems [34]. R ecen tly similar mixing prop erties in an a delic setting hav e b een used in the study of rational p oin ts of gr oup v arieties (see [8], [22], a nd [27]). W e also men tion that fo r the case of group v arieties, the eﬀectiv e coun ting result was obta ined for in tegral p oin ts in [23] and [3 7 ]. W e refer to [31], [26 ], [20], [16], [25 ], [19 ], [36] etc., fo r other t yp es of coun ting and equidistribution results. 1.2. Main results. W e now giv e a precise description of the main results of this pap er. Let K b e a global ﬁeld of c hara cteristic not 2, i.e. a ﬁnite extension of Q or o f F q ( t ) where q is an o dd prime. Let Z b e a symmetric v ariet y in a v ector space V deﬁned ov er K . That is, there exist a connected algebraic almost K - simple group G deﬁned ov er K , a K -represen tation ρ : G → GL ( V ) with ﬁnite k ernel and a non-zero p oint z 0 ∈ V K whose stabilizer H in G is a symmetric K -subgroup of G suc h that Z = z 0 G . By a symmetric K -subgroup of G , we mean a K - subgroup whose iden tity comp onen t coincides with the iden tity comp onen t of the gro up o f ﬁxed p oints G σ for an inv olution σ of G deﬁned o v er K . W e assume tha t the iden tity comp o nent H 0 has no non-trivial K -character. W e ﬁx a basis o f the K - v ector space V K so that one can deﬁne, for an y subring O of K , the subsets V O ⊂ V , Z O ⊂ Z and G O ⊂ G of p oin t s with co eﬃcien ts in O . F o r each place v of K , denote by K v b e the completion of K with respect to the a bsolute v alue | · | v . W e write V v , Z v and G v for V K v , Z K v and G K v , resp ectiv ely . Let S b e a ﬁnite set of places of K con taining all arc himedean (sometimes called inﬁnite) places with G v non-compact. Note that if c har K is p ositive , K do es no t hav e any arc himedean place. W e denote b y O S the ring o f S -integers of S - INTEGRAL POINTS 3 K , that is, O S := { k ∈ K | | k | v ≤ 1 fo r each ﬁnite v 6∈ S } . F or instance, if K = Q , w e ha v e O S = Z S . W e set Z S = Q v ∈ S Z v and similarly G S and H S . Note t ha t t he sets Z O S , G O S and H O S are discrete subsets of Z S , G S , and H S resp ectiv ely , via the diag o nal em b eddings. By a theorem o f Borel and Harish-Chandra in c haracteristic 0 and of Behr and Harder in p ositiv e c haracteristic (see Theorem I.3.2.4 in [35]), the subgroups G O S and H O S are lattices in G S and H S resp ectiv ely . Ag a in, b y a theorem of Bor el, Harish-Chandra, Behr and Harder, the g roup G O S has only ﬁnitely many orbits in Z O S (see Theorem 10 in [21]). Hence our coun ting and equidistribution question of S -in tegra l p oints of Z reduces to coun ting and equidistribution of p oin ts in a single G O S -orbit, sa y , for instance, in z 0 G O S . Set Z S := z 0 G S = Y v ∈ S z 0 G v and let Γ S b e a subgroup of ﬁnite index in G O S . Let µ X S b e a G S -in v ariant measure on X S := Γ S \ G S and µ Y S an H S -in v ariant measure on Y S := (Γ S ∩ H S ) \ H S . F or eac h v ∈ S , w e c ho ose an in v a rian t measure µ Z v on Z v := z 0 G v so that for µ Z S := Q v ∈ S µ Z v , w e ha ve µ X S = µ Z S µ Y S lo cally . F or a subset S 0 ⊂ S , w e set µ Z S 0 = Q v ∈ S 0 µ Z v . F or a Bo r el subset B of Z S , w e set v ol ( B ) := µ Y S ( Y S ) µ X S ( X S ) µ Z S ( B ) . W e assume that G S is non-compact; otherwise Z O S is ﬁnite. By considering a ﬁnite co v ering of G by its simply connected cov er, w e ma y also assume tha t G is simply connected without loss of generality . Before stating our main result, w e summarize our set-up: K is a global ﬁeld suc h that c har( K ) 6 = 2, Z ≃ H \ G is a symmetric v ariet y in a v ector space V deﬁned ov er K where G is an almost K -simple simply -connected K -gro up acting on V suc h that the iden tity comp onen t H 0 has no non-trivial K - c haracter, and S is a ﬁnite set of places of K con taining all the inﬁnite places v with G v non-compact and satisfying that G S is non-compact. Coun ting S -in t egral p oin t s . W e ﬁrst state o ur coun ting results. W e refer to Deﬁnition 6.1 for the notion o f a w ell-r ounded sequence of subsets B n in Z S . Roughly sp eaking, this means that for all small ε > 0, t he b oundaries of B n can b e approximated by neigh b orho o ds whose volume is of ε - order compared to the v olume o f B n uniformly . 4 YVES BENOIST AND HEE OH Theorem 1.1. F or any wel l-r ounde d se quenc e of subsets B n of Z S with volume tending to inﬁnity, we have #( z 0 Γ S ∩ B n ) ∼ v ol( B n ) as n → ∞ . As a corolla ry , we obtain that the num ber of S -integral p oints of size less than T is giv en by the v olume of the corresp onding ball in Z S . A most nat ural wa y to measure the size of an S -integral p oin t is giv en by a height function H S . F or z ∈ Z ( O S ), it is simply H S ( z ) := Y v ∈ S k z k v where the k · k v are no rms on V K v whic h are euclidean when v is inﬁnite and whic h are max norms when v is ﬁnite. This heigh t function H S naturally extends to Z S . Corollary 1.2. As T → ∞ , # { z ∈ z 0 Γ S : H S ( z ) < T } ∼ vol( B S ( T )) . wher e B S ( T ) := { z ∈ Z S : H S ( z ) < T } . When K is a n um b er ﬁeld, Theorem 1.1 is pro v ed with a rate of con vergenc e (see Theorem 12.2). F or instance, we get: Theorem 1.3. L et K b e a numb er ﬁeld. Ther e exists δ > 0 such that as T → ∞ # { z ∈ z 0 Γ S : H S ( z ) < T } = vol( B S ( T ))(1 + O ( T − δ )) . W e will see ( R emark 7.10) that there exist a ∈ Q > 0 , b ∈ Z ≥ 0 and c 1 , c 2 > 0 suc h tha t for T large, c 1 T a log( T ) b ≤ v ol( B S ( T )) ≤ c 2 T a log( T ) b . In general, one cannot c ho ose c 1 = c 2 . The rate of con v ergence in Theorem 1.3 is new ev en for integral p o ints in the generalit y of symmetric v arieties. In this case, as T → ∞ , v ol( B S ( T )) ∼ c T a log( T ) b , for some c > 0. Our pro of o f Theorem 1.3 uses Denef ’s result on lo cal zeta functions whic h is not av ailable in p o sitiv e c har a cteristic. This explains o ur h yp othesis on the c haracteristic o f K . Equidistribution of S -integral p oints . T o motiv ate, consider the case when K = Q and suppo se that Z Z [ p − 1 ] , the set of r a tional p oints in Z with denomina- tors only p o we r of p , is a dense subset in Z R , whic h is often the case. A natural question is when the seque nce of subsets in Z Z [ p − 1 ] consisting of elemen ts of de- nominator precisely p n is equidistributed as n → ∞ . That is, for t w o compact subsets Ω 1 , Ω 2 of Z R , as n → ∞ , { x ∈ Z Q ∩ Ω 1 : p n x ∈ V Z , p ∤ p n x } { x ∈ Z Q ∩ Ω 2 : p n x ∈ V Z , p ∤ p n x } ∼ v ol(Ω 1 ) v ol(Ω 2 ) ? S - INTEGRAL POINTS 5 Once w e note that p n x ∈ V Z is equiv alen t to the condition that t he p -adic maxim um norm of x is at most p n , the ab ov e question can b e rephrased as the question of equidistribution on Z R of the sets { z ∈ Z Z [ p − 1 ] : k z k p = p n } . W e answ er this question in greater generalities: Theorem 1.4. L et S = S 0 ⊔ S 1 b e a p artition o f S . F or a n y wel l-r ounde d se quenc e B n of subsets of Z S 1 with volume tending to inﬁ n ity, and for any c omp act subset Ω ⊂ Z S 0 of p ositive me asur e a n d of b oundary m e asur e 0 , we have # z 0 Γ S ∩ (Ω × B n ) ∼ µ Y S ( Y S ) µ X S ( X S ) µ Z S 0 (Ω) µ Z S 1 ( B n ) as n → ∞ . Note that t he sp ecial case discussed prior to Theorem 1.4 corresp onds to K = Q , S 0 = {∞} , S 1 = { p } , and B n = { z ∈ Z Q p : k z k p = p n } . Note that in all the ab ov e theorems, we may replace z 0 Γ S b y Z O S := Z S ∩ Z O S . as long as w e r enorma lize the volume form so that the v o lume of a subset E ⊂ Z S is giv en b y f v ol ( E ) = X µ Y S ( Y S ) µ X S ( X S ) µ Z S ( E ) (1.5) where w e sum t he contributions from eac h Γ S -orbit in Z O S . Hence w e obta in: Corollary 1.6. Assume S has at le ast two plac es. (1) F or any ﬁnite v ∈ S , the s e ts Z ( T ) := { z ∈ Z O S : k z k v = T } b e c om e e quidistribute d in Z S −{ v } as T → ∞ , subje ct to the c ondition Z ( T ) 6 = ∅ . (2) F or an inﬁnite v ∈ S , the sets Z T := { z ∈ Z O S : k z k v ≤ T } b e c ome e quidistribute d in Z S −{ v } as T → ∞ , pr ovide d Z v is non -c omp act. Again, when K is a n umber ﬁeld, Theorem 1.4 and Corollar y 1.6 are pro ved with a rate of conv ergence (see Corollary 12.3 and Prop osition 13.2 ). F o r instance, w e o btain: Theorem 1.7. L et K b e a numb er ﬁeld and S = S ∞ ⊔ S f b e the p artition of S into inﬁnite and ﬁnite p l a c es. We assume that G S f is non c omp act. Set β T := { z ∈ Z S f : H S f ( z ) < T } . Then ther e e x ist δ > 0 such that for any c omp act subset Ω of Z S ∞ with pie c ewise smo oth b oundary, #( z 0 Γ S ∩ (Ω × β T )) = µ Y S ( Y S ) µ X S ( X S ) w T µ Z ∞ (Ω)(1 + O ( T − δ )) as T → ∞ wher e w T := µ Z S f ( β T ) . Equidistribution of translates of H S -orbits . Set X S = G O S \ G S and Y S = H O S \ H S . Let µ X S and µ Y S b e in v ariant probability measures on X S and Y S resp ectiv ely . The follo wing theorem is a crucial to ol in proving Theorem 1 .1. It states that the translates Y S g is equidistributed in X S as g lea v es compact subsets of H S \ G S . 6 YVES BENOIST AND HEE OH Theorem 1.8. F or any ψ ∈ C c ( X S ) , Z Y S ψ ( y g ) dµ Y S ( y ) → Z X S ψ dµ X S as g tends to inﬁ nity in H S \ G S . The case when K = Q and S = {∞ } , Theorem 1.8 was pro ved in [17], [15], [18] and [45]. In the case when H S is semisimple and non-compact, it is recen tly pro ved in [24], by extending theorems of Mozes-Shah [38] and Dani-Margulis [10 ] in S -a lgebraic settings. None of the ab ov e pap ers address the rate issues , while our pro of giv es eﬀectiv e v ersion in the case when c har ( K ) = 0: a smo oth function on X S is a function whic h is smo oth f or eac h inﬁnite place in S and inv ariant under a compact op en subgroup of G v for each ﬁnite place v ∈ S . The f o llo wing eﬀectiv e v ersion of theorem 1.8 is a crucial to ol in proving Theorem 1.3 as well as other eﬀectiv e r esults in this pap er. Theorem 1.9. F or K numb er ﬁeld, ther e exists κ > 0 such that, fo r any smo oth function ψ on X S with c omp act supp ort, ther e exists c = c ψ > 0 such that     Z Y S ψ ( y g ) dµ Y S ( y ) − Z X S ψ dµ X S     ≤ c H S ( z 0 g ) − κ for al l g ∈ G S . Examples . Let K = Q a nd consider the follo wing pa irs ( V , f ): (A) V : the aﬃne n -space with n ≥ 3 a nd f : an in tegral quadratic form of n -v ariables. If n = 3, w e assume that f do es no t represen t 0 ov er Q . (B) V : the space o f symmetric n × n matrices with n ≥ 3 and f = ± det . (C) V : the space of sk ew-symmetric 2 n × 2 n -matrices with n ≥ 2 and f = ± pﬀaf = ± √ det. F or a p ositiv e integer m , deﬁne V m := { x ∈ V : f ( x ) = m } . Consider the radial pro jection π : V m → V 1 giv en b y x 7→ m 1 /d x where d is the degree o f f . Let V ( Z ) prim b e the set of primitiv e in tegral v ectors in V . F or a ﬁnite set S of primes o f Q con ta ining the inﬁnite prime ∞ , w e denote by h S i ⊂ Q ∗ the m ultiplicativ e semigroup generated b y the ﬁnite primes in S . W e g iv e a partial answ er to the following Linnik problem (see [3 2], [44], [20], [40], [36]): Corollary 1.10. Fix S and ( V , f ) as ab ove. Then ther e e x ist c onstants δ > 0 and ω m , such that fo r any non-e m pty c omp act subse t Ω ⊂ V 1 ( R ) with pie c ewise smo oth b oundary, we have # Ω ∩ π ( V m ( Z ) prim ) = ω m v ol (Ω)(1 + O ( m − δ )) as m → ∞ in h S i , subje ct to V m ( Z ) prim 6 = ∅ . S - INTEGRAL POINTS 7 A sp ecial case of (A) give s a n eﬀectiv e equidistribution for { x ∈ Z 3 : f ( x ) = m } with f = x 2 + y 2 + z 2 or f = x 2 + y 2 − 3 z 2 , hence giving a diﬀeren t pro of of partial cases (b ecause of the restriction on m ) of theorems of Iw aniec [29] and Duke [14]. Note tha t a sp ecial case of (B) give s a n eﬀectiv e equidistribution for the p ositive deﬁnite inte gral matrices of give n determinant. These cases are of sp ecial in t erest since the correspo nding symmetric g r oup H is either compact ov er the reals or a torus. When H is semis imple without compact factors o ver the reals, Corollary 1.10 in its non-eﬀectiv e form, but with no restriction on m , is obta ined in [20] using Ratner’s w ork on the theory o f unip oten t ﬂows. Corollary 1.11. Ke ep the same assumption as in Cor ol lary 1.10 and set m S to b e the pr o duct of the ﬁnite p ∈ S . F or the c ase (A), we f urther supp ose that f r epr esents 0 over Q p for at le ast one p ∈ S . Then ther e exists δ > 0 such that # { x ∈ V ( Z ) prim : k x k ∞ < T , f ( x ) ∈ h S i} = v T (1 + O ( T − δ )) wher e the asymptotic v T is given by the fol low ing sum over the diviso rs m of m d − 1 S : v T = P m | m d − 1 S f v ol ( { x ∈ ( V m ) S : H S ( x ) < T } ) T o prov e corollaries 1.10 a nd 1.11, we will apply the eﬀectiv e v ersions of The- orems 1.1 and 1.4 to V 1 ( Z S ). W e list more examples in section 15. 1.3. Guideline. W e tried to help the reader in writing ”twice ” the pro ofs: In the ﬁrst half of this pap er w e concen trate o n the main term in the counting and equidistribution statemen ts. In the second half, w e follo w t he same strategy but dev elop more tec hnical to ols to obtain the eﬀectiv e v ersions of these statemen ts, i.e. to con trol the error terms. In section 2, we recall the deca y of matr ix co eﬃcien ts for semisimple gro ups G and its a pplicatio n on a homogeneous space Γ \ G of ﬁnite v olume. In section 3, we sho w the w av efron t prop erty for symm etric spaces H \ G ov er lo cal ﬁelds and their pro ducts. In section 4, w e explain how mixing and w av efron t prop erties imply the equidistribution prop erties of translates of H -orbits in Γ \ G give n in Theorem 1.8. In sections 5 and 6, w e explain ho w these equidis tribution prop erties for translates of H -orbit s in Γ \ G allow us to compare for wel l-r ounde d seque nces o f functions on H \ G each sum o ver a Γ-orbit with the integral on H \ G . In section 8, w e giv e examples of w ell-ro unded sequences a nd give pro o fs of Theorem 1.1, Corollary 1.2, Theorem 1.4 and Corollary 1.6. Starting from section 9 , w e prov e the eﬀectiv e results listed in the in tro duction. Theorem 1.9 is pro ved in section 11, and Theorems 1.3, 1.7 , Corollaries 1.10 and 1.11, among other eﬀectiv e applications, are prov ed in section 14. W e list more concrete examples in section 15. In the app endix 16, w e g iv e some general estimates f o r the v olume of balls in the orbits of algebraic groups b o t h ov er the real and p-adic num b ers. 8 YVES BENOIST AND HEE OH W e remark t hat in the whole pap er the assumption of H symmetric is used only to obtain the (eﬀectiv e) w av e front prop ert y for H S \ G S . The metho ds and the argumen t s in this pap er w ork equally w ell for any K -subgroup H with no non-trivial c haracters satisfying the w av e fron t prop erty . Ac knowledgme n t The authors would lik e to thank Alex Goro dnik for helpful con v ersations. 2. The mixing p r oper ty W e ﬁrst recall the Ho we -Mo ore prop erty also called de c ay of matrix c o eﬃcients . Deﬁnition 2.1. A lo c al ly c omp act gr oup G is said to hav e the Howe-Mo or e pr op- erty if, for every unitary r ep r esentation ( H , π ) of G c ontainin g no non-zer o v e c- tors invariant by a normal non-c omp act sub gr oup, we have for al l v , w ∈ H , lim g →∞ h π ( g ) v , w i = 0 . This How e-Mo ore prop ert y is related to the following mixing prop ert y . Let G b e a (unimodula r ) lo cally compact group and Γ a lattice in G , i.e. a discrete subgroup o f ﬁnite cov olume. Let µ X b e a G - in v arian t measure on X := Γ \ G . The group G acts on X by right-translations. Deﬁnition 2.2. The action of G on X is said to b e mixin g if for al l α and β ∈ L 2 ( X ) lim g →∞ Z X α ( g x ) β ( x ) dµ X ( x ) = µ X ( X ) Z X α dµ X Z X β dµ X . The relation b etw een these tw o deﬁnitions is giv en by the follow ing straight- forw ard prop osition. Deﬁnition 2.3. A lattic e Γ in a lo c al ly c omp act gr oup G is c al le d irr e ducible if for any non-c omp act normal s ub gr oup G ′ of G , the sub gr oup Γ G ′ is den se in G . Note that this deﬁnition is sligh tly stronger than the usual deﬁnition since it excludes lattices contained in a prop er subgroup o f G . Prop osition 2.4. L et G b e a lo c al ly c omp act gr oup satisfying the Howe-Mo or e pr op erty and Γ an irr e ducible lattic e in G . Then the action of G on Γ \ G is mixing. Pr o of. This is w ell-kno wn. One ma y assume that α and β b elong to H := L 2 0 ( X ) of square-in tegrable f unctions with zero integral. The G -action by right- translations on H via ( π ( g ) f )( x ) = f ( xg ) is a unitary represen ta tion of G . The irreducibilit y h yp othesis on Γ implies precisely that H do es not contain a n y non- zero v ector in v arian t b y a normal non- compact subgroup of G . Hence, by Deﬁ- nition 2.1, the matrix co eﬃcien ts h π ( g ) α , β i con v erge to 0 as g tends to inﬁnity .  S - INTEGRAL POINTS 9 The main example is due to Ho w e- Mo ore. Theorem 2.5. F or i = 1 , .., m , le t k i b e a lo c al ﬁeld an d G i the gr oup of k i - p oints of a c onne cte d semisimpl e k i -gr oup. Then the pr o duct G := Q m i =1 G i has the Howe-Mo or e pr op erty. In this pap er, “ lo cal ﬁeld” means “ lo cally compact ﬁeld”, i.e. a completion of a global ﬁeld, or, equiv a lently , a ﬁnite extension of R , Q p or F p (( t )). Pr o of. See, for instance, Prop osition I I.2.3 of [35] or [3].  3. The w a vefr ont proper ty The wa v efront prop erty w as intro duced b y Eskin and McMullen for real sym- metric spaces [17]. Let G b e a lo cally compact group and H a closed subgroup of G . Deﬁnition 3.1. The gr oup G has the wavefr ont pr op erty in H \ G if ther e exists a Bor el subset F ⊂ G such that G = H F and, for eve ry neighb orho o d U of e in G , ther e exis ts a neighb orho o d V of e in G such that H V g ⊂ H g U for al l g ∈ F . This prop erty means roughly that the g -translate o f a small neigh b orho o d of the base p oin t z 0 := [ H ] in H \ G remains near z 0 g uniformly ov er g ∈ F . This section is dev oted to proving t he follo wing: Prop osition 3.2. L et k b e a lo c al ﬁeld of char acteristic not 2 , G a c onne cte d semisimple k -gr oup, σ a k -involution of G , G = G k and H a close d sub gr oup of ﬁnite index in the gr oup G σ of σ -ﬁxe d p oints. Then the gr oup G ha s the wavefr ont pr o p erty on H \ G . T o prov e the ab o ve prop osition, we need the f ollo wing tw o lemmas. A k -t o rus S of G is said to b e ( k , σ )- split if it is k -split a nd if σ ( g ) = g − 1 for all g ∈ S . By a theorem of Helminc k and W ang [28], there a re only ﬁnitely man y H -conjugacy classes of maximal ( k , σ )- split tori o f G . Cho ose a set { A i : 1 ≤ i ≤ m } of represen tativ es of H - conj ug acy class of maximal ( k , σ )-split tor i of G and set A = ∪ m i =1 A i ( k ). The following lemma w as pro v ed in [1 7] for k = R , in [4] for a ll lo cal ﬁelds of c haracteristic not 2 (and indep enden tly in [11] when the r esidual c haracteristic is not 2). Lemma 3.3 ( P olar decomposition of symmetric spaces). Ther e e xists a c omp act subset K of G such that G = H AK . The second lemma we need is based on the work of Helminc k and W ang. 10 YVES BENOIST AND HEE OH Let A b e a maximal ( k , σ )-split to rus of G and L the cen tralizer of A in G . The set of ro ots Φ = Φ( G , A ) for the action of A on the Lie algebra of G is a ro ot system. F or eve ry p ositive ro ot system Φ + ⊂ Φ, let N (resp. N − ) b e the unip oten t subgroup of G generated b y the ro ot groups U α (resp. U − α ), for α ∈ Φ + , let P := LN (resp. P − := LN − ) and A + k the W eyl Cham b er : A + k := { a ∈ A k | | α ( a ) | ≤ 1 , for all α ∈ Φ + } . When Φ + v ary , the W eyl c hambers form a ﬁnite cov ering of A k . Since P − = σ ( P ), the pa rab olic k -subgroups P are σ -split, i.e., the pro duct HP is op en in G [28, Prop. 4.6 and 13.4]. Con ve rsely , any minimal σ -split parab olic k -subgroups of G con taining A can b e constructed in t his wa y for a suitable choice of Φ + . Lemma 3.4. (1) The multiplic ation map m : H k × P k → G k is an op en map. (2) Ther e exis ts a b asis o f c omp act neighb orho o ds W of e in P k such that a − 1 W a ⊂ W for al l a ∈ A + k . (3) F or every neighb orho o d U of e in G , ther e exists a neig hb orho o d V of e in G such that H V a ⊂ H aU f o r al l a ∈ A + k . Pr o of. (1) When char( k ) = 0 , it follow s from the fact that Lie algebras of P k and H k generate the Lie algebra of G k as a v ector space. F or a c hara cteristic free argumen t, see [28] or Prop osition I.2.5 .4 in [35]. (2) When char( k ) = 0, note that the action of A + k on the Lie algebra Lie( P k ) giv es a fa mily of commuting semisimple linear maps Ad ( a ) whose eigen v alues ha ve b ounded ab ov e b y 1 in their a bsolute v alues. I t follo ws that there exists a basis of compact neigh b orho o ds W 0 of 0 in Lie( P k ) whic h are inv ariant by all Ad ( a ), a ∈ A + k . It suﬃces to set W = exp ( W 0 ). It is easy to ada pt t his argument in p ositiv e c haracteristic case; write P k = L k N k , and note that L k con tains a A k -in v ariant compact op en subgroup. Now considering the linear group N k as a gr o up of upp er triangular matrices in a suitable basis where elemen ts o f A + k are diagonals with increasing co eﬃcien ts in absolute v alue, we can ﬁnd a basis of compact neighborho o ds W a s desired. (3) Cho ose W as in (2 ) small enough so that W ⊂ U a nd choo se any neigh- b orho o d V of e in G con tained in H W . W e then ha ve H V a ⊂ H W a ⊂ H aU , as required.  Pr o of of Pr op osition 3. 2. W e will prov e that G has the w a ve fron t pro p ert y on H \ G with the subset F = AK deﬁned in Lemma 3.3. By Lemma 3.3, it suﬃces to sho w that for e very neighb orho o d U of e in G , ther e exists a neigh b orho o d V of e in G such that H V g ⊂ H g U for al l g ∈ AK Recall A = ∪ m i =1 A i ( k ) where A i is a maximal ( k , σ )-split torus of G . Fix i and a p ositiv e W eyl cham b er C of A i ( k ). S - INTEGRAL POINTS 11 Since K is a compact set, there exists a neigh b o rho o d U 0 of e in G suc h that k − 1 U 0 k is con ta ined in U fo r all k in K . By Lemma 3.4.(3), there exists a neigh b or ho o d V C of e in G suc h that V C a ⊂ H aU 0 for all a ∈ C . No w set V := ∩ C V C where the inters ection is tak en ov er a ll (ﬁnitely many) p ositiv e W eyl c ham b ers of A i ( k ), 1 ≤ i ≤ m . Then for g = ak ∈ ( ∪ C C ) K = AK with k ∈ K and a ∈ C , w e ha v e H V g ⊂ H V C ak ⊂ H aU 0 k ⊂ H ak U = H g U . This ﬁnishes the pro of. In section 8 , we will use this w av efron t prop erty in the pro duct situation, ow ing to the follo wing straigh tfo rw ard prop osition. Prop osition 3.5. F or i = 1 , . . . , m , let G i b e a lo c al ly c omp act gr oup, H i ⊂ G i a clo s e d sub gr oup, G := Q m i =1 G i and H := Q m i =1 H i . If G i has the wavefr ont pr op erty on H i \ G i for e ach 1 ≤ i ≤ m , then G has the wa vefr ont pr op erty on H \ G . The follo wing t heorem is an immediate consequence of Prop ositions 2.4, 2.5, 3.2 and 3.5. Theorem 3.6. F or i = 1 , .., m , let k i b e a lo c al ﬁeld , G i the g r oup of k i -p oints of a semisimple k i -gr oup, σ i an involution of G i deﬁne d over k i , G σ i i its gr oup of ﬁxe d p oints a n d H i a close d sub gr oup of ﬁnite index of G σ i i . L et G = Q m i =1 G i and H := Q m i =1 H i . Then the gr oup G ha s the wavefr ont pr o p erty on H \ G . Mor e over, for any irr e ducible la ttic e Γ in G , the action of G on Γ \ G is mixing. Note tha t this theorem provides many natural examples of triples ( G, H , Γ ) whic h satisfy the hypothesis o f the prop ositions 4.1, 5.3 and 6.2. 4. Equidistribution of transla tes of H - orbits In this section, let G b e a lo cally compact group, H ⊂ G a closed subgroup, Γ ⊂ G a lattice suc h that Γ H := Γ ∩ H is a lattice in H . Set X = Γ \ G and Y = Γ H \ H . Let µ X and µ Y b e in v ariant measures on X and Y resp ectiv ely . Prop osition 4.1. Supp ose that the a c tion of G on X is mixing and that G has the wavefr ont pr op erty on H \ G . Then the tr anslates Y g b e c ome e quidistribute d in X , as g → ∞ in H \ G . This means tha t as the image of g in H \ G lea v es ev ery compact subsets, the sequence of pro babilit y measures 1 µ Y ( Y ) g ∗ µ Y w eakly conv erges to 1 µ X ( X ) µ X , i.e., for an y ψ ∈ C c ( X ), w e hav e 1 µ Y ( Y ) Z Y ψ ( y g ) dµ Y ( y ) → 1 µ X ( X ) Z X ψ dµ X . (4.2) 12 YVES BENOIST AND HEE OH Pr o of. The f ollo wing pro of is adapted from [17]; we p oin t out that the case when Y is no n-compact requires a bit more care, whic h w as not addressed in [17]. Since G = H F , we ma y assume that g b elongs to the subset F in Deﬁnition 3.1. W e assume, without loss of generalities, that µ X and µ Y are probability measures. Let ψ ∈ C c ( X ). Fix ε > 0. By the uniform con tin uit y of ψ there exists a neigh b or ho o d U of e in G suc h that | ψ ( xu ) − ψ ( x ) | < ε for all u ∈ U and x ∈ X . (4.3) By the w av efron t prop erty of G on H \ G , there exists a compact neigh b or ho o d V ⊂ U of e in G suc h that V g ⊂ H g U for all g ∈ F (4.4) Cho ose a compact subset Y ǫ ⊂ Y of measure at least µ Y ( Y ǫ ) ≥ 1 − ε . Cho ose a Borel subset W ⊂ V in G tr a nsv ersal t o H , i.e., a subset W of G suc h that the multiplication m : H × W → G is inj ective with the image H W b eing an op en neighborho o d of e in G . Using the compactness of Y ǫ and the discreteness of Γ, w e ma y assume that the ima g e of W in H \ G is small enough so that the m ultiplication m : Y ǫ × W → Y ǫ W is a bijection 1 on to its image Y ε W ⊂ X . Let µ W b e the measure on W suc h that µ X = µ Y µ W lo cally . Setting I g := Z Y ψ ( y g ) dµ Y ( y ) , w e need to show that I g → Z X ψ dµ X as g ∈ F go es to inﬁnity in G . (4.5) F or simplicit y , set J g = 1 µ W ( W ) Z Y × W ψ ( y w g ) dµ Y ( y ) dµ W ( w ) and K g = 1 µ W ( W ) Z Y ε W ψ ( xg ) d µ X ( x ) . Roughly sp eaking, we will argue that I g is close to J g as a consequence of the w av efron t prop ert y , J g is close to K g since the v o lume o f Y − Y ε is small, and ﬁnally K g is close to the av erage o f ψ for large g b ecause of the mixing prop erty . 1 When Y is compact, one can choo se the transversal W such that the map Y × W → Y W is bijectiv e onto an op en s ubs e t of X . When Y is not compac t, such a transversal does not alwa ys exist. Here is an ex a mple: let G b e the orthog onal group o f the quadra tic form x 2 + y 2 + z 2 − t 2 on R 4 , v 0 = (1 , 0 , 0 , 0), v 1 = (1 , 0 , 2 , 2), Γ = G Z and H the sta bilizer o f the po in t v 0 . One chec ks ea sily that (a) v 1 = γ v 0 for so me γ ∈ Γ and that (b) v 0 is a limit of elements v 1 h n of the H -orbit of v 1 . Hence there exis ts a sequence g n conv erg ing to e in G such that H g n ∩ γ H 6 = ∅ . T o chec k (a), take γ =     1 0 2 2 0 1 0 0 2 0 1 2 2 0 2 3     . F o r (b), take h n =     1 0 0 0 0 1 0 0 0 0 cosh n − sinh n 0 0 − sinh n co sh n     . S - INTEGRAL POINTS 13 By (4.4), f or eac h w ∈ W and g ∈ F , w e ha ve w g = h g ,w g u for some h g ,w ∈ H . Hence | I g − Z Y ψ ( y w g ) dµ Y ( y ) | = | Z Y ψ ( y g ) d µ Y ( y ) − Z Y ψ ( y h g ,w g u ) dµ Y ( y ) | = | Z Y ( ψ ( y g ) − ψ ( y g u )) dµ Y ( y ) | ≤ ε b y (4.3) . Therefore we ha v e | I g − J g | ≤ ε. By the c hoice o f W , we hav e K g = 1 µ W ( W ) Z W Z Y ε ψ ( y w g ) dµ Y ( y ) dµ W ( w ) and hence | J g − K g | ≤ 2 µ Y ( Y − Y ε ) k ψ k ∞ ≤ 2 k ψ k ∞ ε. Since K g = 1 µ W ( W ) R X ψ ( xg ) 1 Y ε W ( x ) dµ X ( x ) where 1 Y ε W is the c haracteristic function o f W Y ε , the mixing pro p ert y of G on Γ \ G say s that K g con v erges to µ Y ( Y ε ) R X ψ dµ X as g → ∞ in F . Hence f o r g ∈ F larg e enough we hav e, | K g − Z X ψ dµ X | ≤ ε + µ Y ( Y − Y ε ) Z X ψ dµ X ≤ (1 + k ψ k ∞ ) ε. Putting this together, w e g et | I g − Z X ψ dµ X | ≤ | I g − J g | + | J g − K g | + | K g − Z X ψ dµ X | ≤ (2 + 3 k ψ k ∞ ) ε. Since ε > 0 is arbitrary , this show s the claim.  Using Theorem 3.6, w e obtain: Corollary 4.6. L et G , H , Γ b e as in T he or em 3.6. Then the tr anslates Y g := Γ H \ H g b e c ome e quidistribute d in X := Γ \ G as g → ∞ in H \ G . 5. Sums and inte grals Let G b e a lo cally compact g roup, H ⊂ G a closed subgroup, Γ ⊂ G a lat tice suc h tha t Γ H := Γ ∩ H is a lat tice in H . Let x 0 := [Γ] b e the base p oin t in X := Γ \ G , Y = x 0 H and z 0 := [ H ] b e the base p oint in Z := H \ G . W e note that z 0 Γ is a discrete subset of Z . There exist G -in v arian t measures µ X , µ Y and µ Z on X , Y and Z . W e normalize them so that µ X = µ Z µ Y lo cally . F or a giv en sequence of non-negativ e functions ϕ n on Z with compact supp ort, w e deﬁne a function F n on X so that, for x = x 0 g , F n ( x ) is the sum of ϕ n o ver the discrete orbit z 0 Γ g : 14 YVES BENOIST AND HEE OH F n ( x ) := X γ ∈ Γ H \ Γ ϕ n ( z 0 γ g ) for x = Γ g . (5.1) W e w ould lik e to compare the v alues of F n with the space a v erage ov er Z : I n := µ Y ( Y ) µ X ( X ) Z Z ϕ n ( z ) d µ Z ( z ) (5.2) W e remark tha t t his normalized in tegr a l I n do es not dep end on the c hoices of measures. The follo wing prop osition 5.3 say s that the sum F n is asymptotic to the nor- malized in t egr a l I n , a t least w eakly . Prop osition 5.3. Supp ose that the tr anslates Y g b e c ome e quidistribute d in X as g → ∞ in Z . Then f o r any se quenc e of non-ne gative f unction s ϕ n on Z with c omp act supp ort such that max n k ϕ n k ∞ < ∞ and lim n →∞ R Z ϕ n dµ Z = ∞ , the r atios F n ( x ) /I n c onver ge we akly to 1 as n → ∞ . This means that, fo r all α ∈ C c ( X ) , lim n →∞ 1 I n Z X F n ( x ) α ( x ) dµ X ( x ) = Z X α ( x ) dµ X ( x ) . (5.4) Pr o of. Using transitivity prop erties for inv ariant in tegratio n on homogeneous spaces, w e o btain tha t for all α ∈ C c ( X ), Z Γ \ G F n αdµ Γ \ G = Z Γ \ G X γ ∈ Γ H \ Γ ϕ n ( H γ g ) α (Γ g ) dµ Γ \ G (Γ g ) = Z G H \ Γ ϕ n ( H g ) α (Γ g ) d µ Γ \ G H (Γ H g ) = Z H \ G Z Γ H \ H ϕ n ( H g ) α (Γ H hg ) d µ Γ H \ H (Γ H h ) dµ H \ G ( H g ) = Z H \ G ϕ n ( z ) β ( z ) dµ H \ G ( z ) where β is the function on Z giv en b y , β ( H g ) = Z Γ H \ H α (Γ H hg ) d µ Γ H \ H (Γ H h ) = Z Y α ( y g ) dµ Y ( y ) . S - INTEGRAL POINTS 15 By assumption, w e ha v e lim z →∞ β ( z ) = µ Y ( Y ) µ X ( X ) Z X α ( x ) dµ X ( x ) . Since I n = R Z ϕ n → ∞ and ϕ n are uniformly b ounded, b y the dominated con ve r- gence theorem lim n →∞ 1 I n Z Z ϕ n ( z ) β ( z ) dµ Z ( z ) = Z X α dµ X . Hence w e obtain the equality (5.4).  6. Counting and equidistribution W e will now improv e the weak con v ergence in prop osition 5.3 to the p oint wise con v ergence of the functions F n . This requires some hy p othesis on the sequence of functions ϕ n whic h will b e called wel l-r ounde dness . W e kee p the notations of section 5. Deﬁnition 6.1. A se quenc e or a family of non-ne gative inte gr able functions ϕ n of Z with c omp act supp ort is said to b e we l l-r ounde d if for any ε > 0 , ther e exists a neighb orho o d U of e in G , such that the fol lowing holds for al l n . (1 − ε ) Z Z (sup u ∈ U ϕ n ( z u ) ) dµ Z ( z ) ≤ Z Z ϕ n dµ Z ≤ (1 + ε ) Z Z ( inf u ∈ U ϕ n ( z u ) ) dµ Z ( z ) . A se quenc e of subsets B n of Z is said to b e wel l-r ounde d if the se quenc e 1 B n is wel l-r ounde d. Sometimes w e will apply the ab o ve deﬁnition to a contin uous family { ϕ T } of functions or subsets, whose meaning should b e clear. Recall that w e w an t to compare t he orbital sum F n ( x 0 ) = P γ ∈ Γ / Γ H ϕ n ( γ z 0 ) with the a verage I n = µ Y ( Y ) µ X ( X ) R Z ϕ n ( z ) dµ Z ( z ). Prop osition 6.2. K e ep the notations and hyp o thesis of Pr op osition 5.3, and assume that the se quenc e ϕ n is wel l-r ounde d. Then, F n ( x 0 ) ∼ I n as n → ∞ . The notatio n a n ∼ b n means that the ratio of a n and b n tends to 1 a s n → ∞ . Pr o of. Once again, w e ma y normalize the measures so that µ X ( X ) = µ Y ( Y ) = 1. Fix ε > 0 and let U b e a neigh b orho o d of e in G given b y Deﬁnition 6.1. W e in tro duce the functions ϕ ± n on Z deﬁned by ϕ + n ( z ) := sup u ∈ U ϕ n ( z u − 1 ) and ϕ − n ( z ) := inf u ∈ U ϕ n ( z u − 1 ) and their in t egr a ls I ± n := R Z ϕ ± n dµ Z . Note tha t for each n , (1 − ε ) I + n ≤ I n ≤ (1 + ε ) I − n . (6.3) 16 YVES BENOIST AND HEE OH W e also intro duce t he f unctions F ± n on X : F ± n ( x ) = X γ ∈ Γ H \ Γ ϕ ± n ( z 0 γ g ) for x = Γ g . It is easy to ch ec k that, f o r all u ∈ U a nd x ∈ X F − n ( ux ) ≤ F n ( x ) ≤ F + n ( ux ) . Cho ose a no n-negativ e contin uous f unction α on X with R X α = 1 and with supp ort included in x 0 U so that the follow ing holds for all n : Z X α F − n dµ X ≤ F n ( x 0 ) ≤ Z X α F + n dµ X . Applying Prop osition 5.3 to the sequences of functions ϕ ± n , w e obtain, for all n large, (1 − ε ) I − n ≤ F n ( x 0 ) ≤ (1 + ε ) I + n . (6.4) Using the estimations (6 .3) and (6.4), ev ery cluster v alue of the sequence of ratios F n ( x 0 ) /I n is within the interv al [ 1 − ε 1+ ε , 1+ ε 1 − ε ]. Hence this sequence conv erges t o 1.  7. Well-roundedness In this section, w e provide explicit examples of w ell- r o unded sequence s ϕ n in order to apply Prop osition 6.2 . W e start with an observ ation that the pro duct of w ell-rounded sequ ences is ag ain well-rounded. Example 7.1. F o r e ach i = 1 , . . . , m , let G i b e a lo c al ly c omp act gr oup, H i ⊂ G i a close d sub gr oup and ϕ i,n b e a wel l-r ounde d se quenc e o f functions on Z i := H i \ G i . L et G := Q m i =1 G i , H := Q m i =1 H i and Z := Q m i =1 Z i . Then the se quenc e ϕ n deﬁne d b y ϕ n ( z 1 , . . . , z m ) = Y 1 ≤ i ≤ m ϕ i,n ( z i ) is wel l-r ounde d. Pr o of. Fix ε > 0. Let U i b e a neighborho o d of e in G i suc h tha t the functions ϕ ± i,n on Z i deﬁned b y ϕ + i,n ( z i ) = sup u i ∈ U i ϕ i,n ( z i u i ) and ϕ − i,n ( z i ) = inf u i ∈ U i ϕ i,n ( z i u i ) satisfy (1 − ε ) R Z i ϕ + i,n ≤ R Z i ϕ i,n ≤ (1 + ε ) R Z i ϕ − i,n . Let U := Q U i , a nd ϕ ± n b e t he f unctions on Z deﬁned b y ϕ + n ( z ) = sup u ∈ U ϕ n ( z u ) and ϕ − n ( z ) = inf u ∈ U ϕ n ( z u ) so that ϕ ± n = Q ϕ ± i,n and (1 − ε ) m R Z ϕ + n ≤ R Z ϕ n ≤ (1 + ε ) m R Z ϕ − n . Hence the sequence ϕ n is w ell-rounded.  S - INTEGRAL POINTS 17 The next example deals with the constan t seque nces. It will b e used bo th for the arc himedean and the no n-arc himedean factors. Example 7.2. L et G b e a lo c al ly c omp act gr oup, H a close d sub gr oup of G , Z = H \ G , and ϕ ∈ C c ( Z ) with ϕ ≥ 0 and ϕ 6 = 0 . Then the c onstant se quenc e ϕ n = ϕ is wel l-r ounde d. Pr o of. Use the uniform contin uit y of ϕ and the compactness of its supp ort.  The f o llo wing example deals with the arc himedean fa ctors. Example 7.3. L et G b e a r e al semisimple Lie gr oup with ﬁnitely m a n y c onne cte d c omp onents, V a ﬁnite di m ensional r epr esentation of G , Z a close d G -orbit in V with an invariant me asur e µ and k . k an euclide an norm on V . Then the family of b al l s B T := { z ∈ Z | k z k ≤ T } , T ≫ 1 is w el l-r ounde d. Pr o of. By Corolla ry 16.3 .a o f the a pp endix, w e ha v e µ ( B T ) ∼ T c T a (log T ) b for some a ∈ Q ≥ 0 , b ∈ Z ≥ 0 and c > 0. It is easy to deduce t he claim from the ab ov e asymptotic using the assumption that the action of G is linear on Z .  As for the non-arc himedean factors, we hav e: Example 7.4. L et k b e a non-ar chime de an lo c al ﬁ eld, G the gr oup of k -p oints of a c onne cte d semisimple k -gr oup, ρ : G → GL ( V ) a r epr esentation of G deﬁne d over k , Z a clos e d G -orbit in V w i th an invariant me asur e and k · k a norm on V . Then , b oth the fami ly o f non empty b al ls B T := { z ∈ Z | k z k ≤ T } , and the family of non- e mpty spher es S T := { z ∈ Z | k z k = T } , ar e wel l-r ounde d. Pr o of. Since the a ction of G on V is linear, the stabilizer in G of the norm is a compact op en subgroup of G . Hence this example is a special case of the following easy assertion.  Example 7.5. L et G b e a lo c al ly c omp act (unimo dular) gr oup, H a close d (uni- mo dular) sub gr oup o f G , Z = H \ G and U a c omp act op en sub gr oup of G . Then any se quenc e ϕ n of non -ne gative U -inva ri a nt L 1 -functions on Z is wel l-r ounde d. As the last example, w e will sho w that a sequence of the heigh t balls is w ell rounded. W e will need the following basic lemma. Lemma 7.6. L et I b e a ﬁni te set. F or e ach i ∈ I , let τ i > 1 and λ i > 0 b e given. L et λ : N I → R + b e given by λ ( m ) = P i ∈ I λ i m i for m = ( m i ) , and P : N I → R + a function giv en by P ( m ) = Q i ∈ I P i ( m i ) wher e P i is a r e al-value d function of a variable x given by a p olynomia l e x pr ession in ( x, τ x i ) and which is p ositive on N . Then we ha ve w t +1 = O ( w t ) for t lar ge 18 YVES BENOIST AND HEE OH wher e w t := X { m ∈ N I , λ ( m ) ≤ t } P ( m ) . Pr o of. Since eac h P i is p ositiv e on N , there exists C > 0 suc h that for i ∈ I and n ∈ N , one has P i ( n + 1) ≤ C P i ( n ) . Hence for eac h m ∈ N I and eac h e in the basis E of N I , o ne ha s P ( m + e ) ≤ C P ( m ) . Setting t 0 := min i λ i = min e ∈ E λ ( e ), one gets w t + t 0 ≤ X e ∈ E X λ ( m ) ≤ t P ( m + e ) ≤ r C w t . for r = | I | . Hence we conclude that w t +1 ≤ ( r C ) k w t with k = 1 t 0 + 1.  Remark 7.7. One can impro v e the conclusion of Lemma 7.6: ther e exist a ≥ 0 , b ∈ Z ≥ 0 and c 1 , c 2 > 0 such that, for t lar g e , c 1 e at t b ≤ w t ≤ c 2 e at t b . Mor e over, setting C i τ d i x i x b i for the dominant term of P i ( x ) , the ex p onents a and b ar e given r esp e ctively by e a = max i ∈ I τ d i /λ i i and b is give n by b = P i ( b i + 1) − 1 wh er e the sum is taken ove r al l i such that e a = τ d i /λ i i . The pro of is a straigh tforward induction on | I | . Here is a sk etc h: one ma y assume that, for all i , P i ( x ) = C i e a i x x b i and λ i = 1. One ﬁxes i 0 ∈ I , set ˇ I := I −{ i 0 } and writes w t := P 1 ≤ n ≤ t P i 0 ( n ) ˇ w t − n where, b y induction h yp othesis, ˇ w t satisﬁes a similar estimation, a s t → ∞ : ˇ c 1 e ˇ at t ˇ b ≤ ˇ w t ≤ ˇ c 2 e ˇ at t ˇ b for some ˇ a ≥ 0, ˇ b ∈ Z ≥ 0 and ˇ c 1 , ˇ c 2 > 0. F rom that, one gets the required estimation fo r w t . F or the r est of this section, let I be a ﬁnite set. F or each i ∈ I , let k i b e a lo cal ﬁeld of characteristic 0, G i the g roup of k i -p oints of an a lgebraic k i -group, V i an algebraic represen tation of G i , and Z i ⊂ V i a non-zero closed G i -orbit with an inv ariant measure µ i . W e set G := Y i ∈ I G i , Z := Y i ∈ I Z i , µ := ⊗ i ∈ I µ i . Let I ∞ ⊂ I b e the set o f indices with k i arc himedean, and I f := I \ I ∞ . The partition I = I ∞ ⊔ I f induces decomp ositions G = G ∞ × G f of the gr oup, Z = Z ∞ × Z f of the orbit, and µ = µ ∞ ⊗ µ f of the in v ariant measure. S - INTEGRAL POINTS 19 Let k · k i b e a norm on V i . W e a ssume that k · k i is euclidean if i ∈ I ∞ and a max norm otherwise. These norms deﬁne a heigh t function h : Z → R + h ( z ) = Y i ∈ I k z i k i . Since eac h Z i is a closed non-zero subset in V i , w e hav e min z ∈ Z i k z k i > 0 and hence h is a prop er function on Z . W e can a lso write h = h ∞ ⊗ h f where h ∞ := Q i ∈ I ∞ k · k i and h f := Q i ∈ I f k · k i . Set b T := { z ∈ Z ∞ | h ∞ ( z ) ≤ T } , β T := { z ∈ Z f | h f ( z ) ≤ T } , V T := µ ( B T ) , v T = µ ∞ ( b T ) a nd w T = µ f ( β T ) . Lemma 7.8. Assume that h ∞ is not c onstant on Z ∞ . (1) Ther e exis t a ∈ Q ≥ 0 , b ∈ Z ≥ 0 , and c > 0 such that as T → ∞ , v T ∼ c T a (log T ) b and d dT v T ∼ c d dT ( T a (log T ) b ) . (2) Ther e exis t c onstants κ > 0 and C 1 > 0 such that for al l ε ∈ ]0 , 1[ and al l T ≥ 0 v (1+ ε ) T − v T ≤ C 1 ( v T + 1) ε κ . (3) F or T l a r ge, one has w 2 T = O ( w T ) . (4) Ther e exis t κ > 0 such that uniformly for T lar ge a n d ε ∈ ]0 , 1 [ , V (1+ ε ) T = (1 + O ( ε κ )) V T . Pr o of. (1): Apply Prop osition 16 .2 of the app endix to the regular function F := h 2 ∞ on the orbit Z ∞ . Note that since v T is an increasing function of T , one has a ≥ 0 . Moreo ver, note that, when a = b = 0, the o rbit is o f ﬁnite volume and hence compact. (2): First note that, since h ∞ is not constan t on Z ∞ , the function v T is con- tin uous. When T is large, w e use ( 1 ) to get the following b ound v (1+ ε ) T − v T = O ( εv T ) whic h is unifor m in ε ∈ ]0 , 1[. When T is b ounded, we use the fa ct that the function v T is diﬀerentiable except at the critical v alues τ of h ∞ . Since h 2 ∞ is a regular function, there are only ﬁnitely man y suc h critical v alues τ . Around these p oin t s, there exists a constan t κ , 0 < κ < 1 suc h tha t , fo r ε > 0 small, one has the follo wing b ound for the deriv a tiv e: v ′ τ ± ε = O ( ε κ − 1 ) . (7.9) This assertion is a consequence o f Theorem 16.1 o f the app endix. More precisely , set f := ± ( h ∞ − τ ). Since Z ∞ is smo oth, one can choose ε 0 > 0 and an op en co ve r ing U j of f − 1 (] − ε 0 , ε 0 [) b y op en sets bianalytically homeomorphic to balls. A partition of unit y giv es us C ∞ functions ϕ j with compact supp o rt in U j suc h 20 YVES BENOIST AND HEE OH that P j ϕ j = 1 o n f − 1 (] − ε 0 / 2 , ε 0 / 2[). W e simply apply Theorem 16.1 to these functions f and ϕ j to get (7.9). In tegrating v ′ t on t he in terv al [ T , (1 + ε ) T ], and using (7.9) near the critical v alues in this in terv al, one gets, uniformly for ε small and T b ounded, v (1+ ε ) T − v T = O ( ε κ ) . Putting these together pro ves the claim. (3): W e will assume, as w e may , t hat inf z ∈ Z i k z k i ≥ 1 for eac h i ∈ I f . F or any | I f | -tuple m = ( m i ) i ∈ I f ∈ N I f , w e set S ( m ) = Y i ∈ I f S i ( m i ) where S i ( m i ) := { z ∈ Z i : k z k i = m i } . Letting ω m := µ f ( S ( m )) and π m := Q i m i , o ne ha s w T = P { m ∈M ,π m ≤ T } ω m where M ⊂ N I f consists of m ∈ N I f with non-empt y S ( m ). The main p oin t o f the pro of is to use the form ula for ω m giv en b y Theorem 16 .6 of the app endix. According to this formula, there is a ﬁnite partit ion of M in ﬁnitely many pieces M α suc h tha t - each piece M α is a pro duct of subsets M α,i of N whic h are either p oin ts or of the form { m i = c α,i q n i α,i : n i ∈ N } for some p ositiv e in tegers c α,i , q α,i , - on each piece M α , the volume ω m is giv en by a formu la Q i ∈ I f P α,i ( n i , q n i /d α,i ) where P α,i is a p olynomial and d a p o sitiv e integer. According to Lemma 7.6 with T = 2 t , the v olume w α,T := P { m ∈M α ,π m ≤ T } ω m satisfy the b ound w α, 2 T = O ( w α,T ). Hence one ha s w 2 T = O ( w T ) a s required. (4): Let T 0 := inf z ∈ Z ∞ h ∞ ( z ) > 0. According to (2), there exists C > 0 such that for T la rge V (1+ ε ) T − V T = P m ( v (1+ ε ) T /π m − v T /π m ) ω m ≤ C ε κ  P m v T /π m ω m + P m ω m  ≤ C ε κ ( V T + w 2 T /T 0 ) where the a b o ve sums are o ver all the m ulti- indices m ∈ N I f with T 0 π m ≤ 2 T . Then, applying (3) t wice, there exists C ′ > 0 suc h that for T large V (1+ ε ) T − V T ≤ C ε κ ( V T + C ′ w T / 2 T 0 ) ≤ C (1 + C ′ v − 1 2 T 0 ) ε κ V T , as required, since v 2 T 0 > 0.  S - INTEGRAL POINTS 21 Remark 7.10. One has the follo wing estimate for the v olume V T of the height ball : ther e exis t a ∈ Q ≥ 0 , b ∈ Z ≥ 0 and c 1 , c 2 > 0 such that for T lar ge, c 1 T a log( T ) b ≤ V T ≤ c 2 T a log( T ) b . This is a stra ig h tforward consequence of the formula V T := R ∞ 0 w T /t v ′ t dt and of the estimation of w T and v ′ T giv en in R emark 7 .7 a nd Lemma 7.8 (1) . Prop osition 7.11. (Height ball) The fa m ily of height b al ls B T := { z ∈ Z | h ( z ) ≤ T } , T ≫ 1 , is wel l r ounde d. Pr o of. W e will assume a s w e ma y that all the orbits Z i ha ve p ositiv e dimension. When I ∞ = ∅ , the w ell-roundedness of B T is a consequence of Example 7 .5. Hence w e will assume that I ∞ 6 = ∅ . When the heigh t function h ∞ is constan t on Z ∞ , the w ell-roundedness of B T is a consequence of Example 7.1. When the height function h ∞ is no t constant on Z ∞ , the w ell-roundedness of B T follo ws from Lemma 7.8 (4) and of the linearit y of the action of eac h G i on V i .  Although w e stat ed the ab ov e prop o sition o nly for c haracteristic 0 ﬁelds, when all the k i ha ve p ositive c haracteristic, the heigh t balls are also w ell-rounded by Example 7.5. 8. Applica tions W e will b e a pplying t he following theorem a nd corollary to the ab ov e examples of w ell ro unded sequences. Theorem 8.1. L et I b e a ﬁnite set. F or e ach i ∈ I , let k i b e a lo c al ﬁeld of char acteristic not 2 , G i the gr oup of k i -p oints of a s e m isimple algebr aic k i - gr oup, H i ⊂ G i the k i -p oints of a symmetric k i -sub gr oup. Set G I := Q i ∈ I G i , H I := Q i ∈ I H i , Z I := H I \ G I and z 0 = [ H I ] . L et Γ b e an irr e ducible lattic e of G I such that Γ H := Γ ∩ H I is a lattic e in H I . Then for any se quenc e B n of wel l-r ounde d subse ts of Z I with volume tending to inﬁnity, we have, as n → ∞ , #( z 0 Γ ∩ B n ) ∼ µ Y ( Y ) µ X ( X ) µ Z I ( B n ) , wher e X = Γ \ G I , Y = Γ H \ H I and the volumes ar e c ompute d using i n variant me asur es as in (5.2). Pr o of. Use Corollary 4 .6 a nd Prop osition 6.2 with ϕ n := 1 B n .  In the pro duct situation of Z I = Z I 0 × Z I 1 , w e will b e taking a w ell-rounded sequence s of Z I whic h are pro ducts of a ﬁxed compact subset in o ne factor Z I 1 and a we ll-rounded sequence of subsets in the other factor Z I 0 . This will giv e us equidistribution results in t he space Z I 1 when Z I 0 is non-compact. 22 YVES BENOIST AND HEE OH Corollary 8.2. Ke eping the same hyp othesis as in The or em 8.1, let I = I 0 ⊔ I 1 b e a p artition of I . L etting B n b e a wel l-r ounde d se quenc e of s ubsets of Z I 1 with volume g o ing to inﬁnity, c onsider the fol lowing discr ete multisets Z ( n ) of Z I 0 : Z ( n ) := { z ∈ Z I 0 | ( z , z ′ ) ∈ z 0 Γ ∩ ( Z I 0 × B n ) for som e z ′ ∈ Z I 1 } . Then, as n → ∞ , the s ets Z ( n ) b e c ome e quidistribute d in Z I 0 with r esp e ct to a suitably n ormalize d i n variant me asur e. In fact, f o r any ϕ ∈ C c ( Z I 0 ) , lim n →∞ 1 µ Z I 1 ( B n ) X z ∈ Z ( n ) ϕ ( z ) = µ Y ( Y ) µ X ( X ) Z Z I 0 ϕ dµ Z I 0 . In p articular, Z ( n ) is non-empty for al l lar ge n . Multiset means that the p oints of Z ( n ) ar e coun ted with m ultiplicit y according to the cardinality o f t he ﬁb ers of the pro jection z 0 Γ ∩ ( Z I 0 × B n ) → Z ( n ). Since z 0 Γ is discrete and B n is relat ively compact, w e note that these ﬁb ers are ﬁnite and that Z ( n ) is discrete in Z I 0 . Pr o of. It suﬃces to pro ve the claim f o r non-negativ e functions ϕ ∈ C c ( Z I 0 ). Deﬁne a sequenc e of f unctions ϕ n on Z b y ϕ n ( z , z ′ ) := ( ϕ ⊗ 1 B n )( z , z ′ ) = ϕ ( z ) 1 B n ( z ′ ) f or ( z , z ′ ) ∈ Z I 0 × Z I 1 . By Example 7.1, this sequence ϕ n is w ell-rounded and X z ∈ Z ( n ) ϕ ( z ) = X z ∈ z 0 Γ ϕ n ( z ) since Z ( n ) is a m ultiset. By Corollary 4.6, we can apply Prop osition 6.2 to the sequence ϕ n and obtain lim n →∞ 1 µ Z I 1 ( B n ) X z ∈ Z ( n ) ϕ ( z ) = lim n →∞ 1 µ Z I 1 ( B n ) X z ∈ z 0 Γ ϕ n ( z ) = µ Y ( Y ) µ X ( X ) Z Z I 0 ϕ dµ Z I 0 .  Remark 8.3. In Prop ositions 5.3 and 6.2, o ne can replace the h yp othesis “the L 1 -norm of ϕ n go es to inﬁnity ” b y the h yp ot hesis that “the supp ort of ϕ n is non-empt y and go es to inﬁnity” i.e. for ev ery compact C of Z , ϕ n | C is n ull for all n large. The pro of is exactly the same. A similar remark applies to Theorem 1.1, 8.1 a nd Corolla r y 8.2. This remark is useful fo r the non- empt y spheres in Example 7.4, since it a v o ids to che c k that their volume g o es to inﬁnity with the radius. Pr o of of The or ems 1.1, 1.4, 1.8 and Cor ol la ry 1.2. W e ar e no w ready to pr ov e the non-eﬀectiv e stat ements in the in tro duction. Theorem 1.1 is a consequence of Theorem 8 .1 with I = S , G S = G S , H S = H S and Γ = Γ S . The only thing we hav e to c heck is that Γ S is an irreducible lattice in G S . This is the following classical lemma 8.4. S - INTEGRAL POINTS 23 Theorem 1.4 is a consequence of Corollary 8.2 with I = S and I 0 = S 0 . Note that the pro jection z 0 Γ → Z I 0 is injectiv e and hence the multise t Z ( n ) is a set. Theorem 1.8 is an immediate consequence of Corollary 4.6. Corollary 1.2 is a consequence of Theorem 1.1 and Prop osition 7.1 1.  Let G b e a connected semisim ple group deﬁned ov er a global ﬁeld K , and let S b e a ﬁnite set o f places of K con taining all archime dean places v suc h that G v is non-compact. Recall that these conditions a ssure that the subgroup G O S is a lattice in G S := Q v ∈ S G v . Lemma 8.4. L et Γ S b e a sub gr oup of ﬁnite index in G O S . S upp ose that G is simply c onne cte d, almost K -sim ple and that G S is non-c omp act. Then Γ S is an irr e ducible lattic e in G S (se e Deﬁnition 2.3). Pr o of. Since G is simply connected and G S is non-compact, then G has the strong appro ximation prop ert y with resp ect to S , that is, the diagona l em b edding o f G K is dense in the S -a deles G A S , i.e., the adeles without S -comp o nen t (see [42, Th. 7.12] for characteristic 0 cases a nd [41] for the p ositiv e c haracteristic case). Since G is K -simple, it follows that Γ S is an irreducible lattice in G S [35, Cor. I.2.3.2 & Th. I I.6.8].  F or the rest of this pap er, w e will transform the pro ofs explained in the ab o ve c hapters into eﬀectiv e pro of s. F or that w e need to control precisely all the error terms app earing in these pro ofs. There are mainly four error terms t o control. The ﬁrst three come from the mixing pr o p ert y , t he wa v e f r o n t prop erty and the approximation of µ Y b y a smo oth function. Their con trol will give the equidistribu- tion sp eed of the translates of µ Y . The last error t erm comes fr o m the we ll ro undedness of the balls B T . W e will dedicate one section to eac h of these terms. 9. Effective mixing In this section, w e in tro duce nota tions whic h will b e used through the section 14 and w e describ e a n eﬀectiv e v ersion (Theorem 9.2) o f the mixing prop erty based on the uniform deca y of ma t rix co eﬃcien ts. W e let K b e a n umber ﬁeld, G a connected simply connected almost K -simple group a nd H a K -subgroup o f G with no non-trivial K -c haracter. Let S b e a ﬁnite set of places of K con taining all the inﬁnite places v suc h that G v is non-compact. W e write S ∞ and S f for t he sets of inﬁnite and ﬁnite places in S resp ectiv ely . W e a ssume that G S := Q v ∈ S G v is non-compact. Let Γ S b e a ﬁnite index subgroup of G ( O S ). Note that H S ∩ Γ S is a lattice in H S . 24 YVES BENOIST AND HEE OH Set X S := Γ S \ G S and Y S = Γ S ∩ H S \ H S . L et µ X S and µ Y S denote the in v arian t probabilit y measures on X S and Y S resp ectiv ely . Se t Z S := H S \ G S . F or eac h v ∈ S , c ho o se an in v ariant measure µ Z v on H v \ G v so that the inv ariant measure µ Z S := Q v ∈ S µ Z v on Z S satisﬁes µ X S = µ Y S µ Z S lo cally . F or S 0 ⊂ S , w e set µ Z S 0 := Q v ∈ S 0 µ Z v . By a smo oth function on X S w e mean a function whic h is smo ot h on eac h G ∞ -orbit and whic h is inv ariant under a compact op en subgroup of G f . The notation C ∞ c (Γ S \ G S ) denotes the set of smo oth functions with compact supp ort on G S . F or eac h v ∈ S , recall the “ Cartan” decomp osition due to Bruhat and Tits in [5] and [6]: one has G v = M v Ω v B + v M v where M v is a go o d ma ximal compact subgroup, B + v a p ositiv e W eyl c hamber of a maximal K v -split torus and Ω v is a ﬁnite subset in the cen tralizer of B v . F or simplicit y , w e set G ∞ = G S ∞ and G f = G S f . W e also set M ∞ := Q v ∈ S ∞ M v and M f := Q v ∈ S f M v . Let X 1 , · · · , X d b e an ortho normal basis of the Lie algebra of M ∞ with resp ect to an Ad-inv ariant scalar pro duct. W e denote b y D the elliptic op erator D := 1 − P d i =1 X 2 i . Fix any closed em b edding of Z = H \ G in to a ﬁnite dimensional v ector space V deﬁned ov er K ; suc h an em b edding alw ays exists b y the we ll kno wn theorem of Chev alley . T o measure ho w fa r an elemen t z ∈ Z S is from t he base p oint z 0 = [ H S ] in Z S := H S \ G S , we may use a height function (9.1) H S ( z ) := Y v ∈ S k z v k v where k · k v is a norm on V v . This norm is assumed to b e euclidean when v is an inﬁnite place and a max norm when v is a ﬁnite place. Note that the heigh t function H S : Z S → R + is a prop er function. Theorem 9.2. Ther e exists κ > 0 and m ∈ N such that for any op en c om- p act sub gr oup U f of G f , ther e exists C U f > 0 satisfying that for a n y ψ 1 , ψ 2 ∈ C ∞ c ( X S ) U f and a n y g ∈ G S ,     h g ψ 1 , ψ 2 i − Z X S ψ 1 dµ X S Z X S ψ 2 dµ X S     ≤ C U f H S ( z 0 g ) − κ kD m ( ψ 1 ) k L 2 kD m ( ψ 2 ) k L 2 . Pr o of. The ab ov e claim is a straightforw ard consequence o f Theorem 2 .20 of [22 ] based on the results of [7] and [39]. [22 , Theorem 2.20] relies on the fo llo wing h yp othesis: ”the only c ha r acter app earing in L 2 (Γ S \ G S ) is the trivial one”. This h yp othesis is satisﬁed here since the non compactness of G S and the simply- connectednes s of G imply the irreducibility of Γ S b y Lemma 8.4 . The conclusion of [22, Theorem 2.20] is the ab ov e claim where H − κ S is replaced b y a function e ξ G whic h is a pro duct o ver v ∈ S o f bi- M v -in v ariant functions ξ ′ v S - INTEGRAL POINTS 25 satisfying ξ ′ v ( a ) ≤ Y α ∈ Q v α ( a ) − 1 / 2+ ε for all a ∈ B + v where Q v is a maximal strongly orthogonal system of t he ro ot system of ( G v , B v ) W e only hav e to c hec k that this function e ξ G is b ounded by a m ultiple of H − κ S . F or that, denote by ρ the represen tation of G in to GL ( V ) suc h tha t the stabilizer of z 0 ∈ V K is H a nd c ho ose a w eight λ larger on B + v than an y w eigh t of ρ . Then there exists a p ositiv e in teger k suc h that, for all a ∈ B + v , k z 0 ρ ( a ) k v ≤ k z 0 k v k ρ ( a ) k v ≤ k z 0 k v | λ ( a ) | v ≤ k z 0 k v Y α ∈ Q v α ( a ) k . Since M v and Ω v are compact subsets , b y the contin uit y , this implies that there exists κ > 0 and c > 0 suc h that ξ ′ v ( g ) ≤ c k z 0 ρ ( g ) k − κ v for all g ∈ G v . This implies our claim.  10. Injective radius and the appro xima tion by smooth functions The aim o f this section is to g et an eﬀectiv e upp er b ound on the v olume of the set of p oints in Y S with small injectivit y radius in X S and appro ximate the c haracteristic f unction Fix a closed em b edding G ֒ → GL N . W e ma y consider eac h elemen t g of G S as an | S | -tuples of N × N matrices g v . W e a lso ﬁx a norm k . k v on eac h of these K v -v ector spaces of matrices. F or x ∈ X S , consider t he pro jection map p x : G S → X S giv en b y g 7→ xg . The injectivit y radius r x is deﬁned to b e r x = sup { r > 0 : p x | B r × M f is injectiv e } where B r = { g ∈ G ∞ : max v ∈ S ∞ k g v − e k v ≤ r } . Of course, this deﬁnition mak es sense only when S ∞ is no n- empt y . This do es not matter since, when S ∞ is empt y , X S is compact. Lemma 10.1. Supp ose S ∞ 6 = ∅ . F or any x ∈ X S , one has r x > 0 . Pr o of. Since Γ S do es not meet G f and G f is no r ma l in G S , the group G f acts freely on X S . Hence p z | { e }× M f is injectiv e. Since M f is compact a nd p x is lo cally injectiv e, p x | B r × M f is still injectiv e for some small r > 0.  Moreo ve r w e ha ve a quan titativ e v ersion of t he ab o v e lemma. Lemma 10.2. Supp o s e S ∞ 6 = ∅ . Ther e exist c 1 > 0 , p 1 > 0 such that fo r al l suﬃciently smal l ε > 0 , µ Y S ( { y ∈ Y S | r y < ε } ) ≤ c 1 ε p 1 . 26 YVES BENOIST AND HEE OH Pr o of. W e use the reduction theory for H S (cf. [42]). W e ﬁrst recall what a Siegel set is. Let A b e a maximal K -split torus of H and P a minimal parab olic subgroup con taining A . Then P = NRA where R is a Q -anisotro pic reductiv e subgroup and N the unip oten t radical of P . Set A ∞ := Q v ∈ S ∞ A ( K v ) a nd simi- larly N ∞ and R ∞ . Denoting by ∆ the system of simple ro ots of H ∞ determined b y the choic e of P , we set for t > 0, A t = { a ∈ A ∞ : α ( a ) ≥ t for all α ∈ ∆ } . Then for a compact subset ω ⊂ N ∞ R ∞ and a ma ximal compact subgroup K 0 of H S , the set Σ t := ω A t K 0 is called a Siegel set. Now t he reduction theory sa ys that there exist h 1 , · · · , h r ∈ H S , and a Siegel set Σ t 0 = ω A t 0 K 0 suc h tha t H S = ∪ r i =1 ( H S ∩ Γ S ) h i Σ t 0 . Let M ′ f := ∪ r i =1 h i M f h − 1 i . As in Lemma 10.1, there exists ε 0 > 0 suc h that (10.3) Γ S ∩ B ε 0 M ′ f = { e } . Set C ε := ∪ r i =1 h i { w ak ∈ Σ 0 : t 0 ≤ α ( a ) ≤ ε − r 0 for eac h α ∈ ∆ } , where r 0 > 0 is chose n indep enden t of ε , so that, f or all g in C ε and v ∈ S ∞ , o ne has k g v k v ≤ ε − 1 / 4 and k g − 1 v k v ≤ ε − 1 / 4 . Let Y ′ ε denote the image of C ε in Y S under the pro jection H S → Y S . The in tegratio n for m ula [42, p. 2 13] sho ws that for some constant c 1 > 0 and p 1 > 0, µ Y S ( Y S − Y ′ ε ) ≤ c 1 ε p 1 . Hence it is enough to sho w that for all z ∈ Y ′ ε , one has r z ≥ ε . Suppose p z ( x ) = p z ( y ) with x, y ∈ B ε M f and write z = Γ S g for some g ∈ C ε . W e w ant to pro v e that x = y . The elemen t γ := g xy − 1 g − 1 b elongs to Γ S . Moreo v er, for some ﬁxed constant c > 1 , one has, for a ll v ∈ S ∞ , k γ − e k v = k g v ( x v − y v ) y − 1 v g − 1 v k v ≤ c ε − 1 / 2 k x v − y v k v k y − 1 v k v ≤ c 2 ε 1 / 2 . But the ﬁnite comp onen t of γ is in M ′ f , hence, γ is in B c 2 ε 1 / 2 × M f and one gets from (10.3) that, for ε < c − 4 ε 2 0 , one ha s γ = e. Therefore x = y as w ell.  F or all v ∈ S , we c ho ose a small neigh b orho o d s v of 0 in a supplemen tary subspace of the Lie algebra h v in g v and set s := Q v ∈ S s v . The set W := exp ( s ) is then a tra nsve rsal to H S in G S . W e set µ W the measure on W suc h that dµ X S = dµ Y S dµ W lo cally . S - INTEGRAL POINTS 27 Recall that B ε denotes the ball of cen ter e and radius ε in G ∞ and let U ε b e the ball of cen ter e and radius ε in G f : B ε = { g ∈ G ∞ | max s ∈ S ∞ k g − e k v ≤ ε } , (10.4) U ε = { g ∈ G f | max s ∈ S f k g − e k v ≤ ε } . (10.5) W e ﬁx ε 0 > 0 small. F or ε small w e let H ε := H S ∩ B ε U ε 0 and W ε := W ∩ B ε U ε 0 so tha t t he m ultiplication H ε × W ε → H ε W ε is an homeomorphism onto a neigh- b orho o d of e . Fix m > dim G ∞ and κ > 0 satisfying Theorem 9.2 and ﬁx l ∈ N as in Lemma 11.3. W e can assume that U lε 0 ⊂ U f ∩ M f and let U ′ f b e an op en subgroup of G f suc h tha t U ε 0 U ′ f = U ε 0 . Lemma 10.6. L et Y ε := { y ∈ Y S | the map g 7→ y g is injective on H ε W ε } . Ther e exis t c 1 > 0 and p 1 > 0 such that for al l smal l ε > 0 , µ Y S ( Y S − Y ε ) ≤ c 1 ε p 1 . Pr o of. When S ∞ 6 = ∅ , since B ε B ε ⊂ B ε 1 / 2 for ε small, the set Y ε con tains the set of p oin t s y suc h that r y ≥ ε 1 / 2 . Just apply then Lemma 10.2. When S ∞ = ∅ , X S is compact, hence Y ε is equal to Y S for ε 0 and ε small.  The fo llowing prop osition pro vides the appro ximation of the c haracteristic function 1 Y ε W b y a smo oth function ϕ ε with the con trolled Sob olev norm. W e ﬁrst recall the Sob olev norm S m ( ψ ) of a f unction ψ ∈ C ∞ c ( X S ). Cho o se a basis X 1 , ..., X n of the L ie algebra of G ∞ . F or eac h k -tuple of intege rs a := ( a 1 , ..., a k ) with 1 ≤ a i ≤ n , the pro duct X a := X a 1 . . . X a k deﬁnes a left-in v ariant diﬀeren tial op erat o r on G S , hence a diﬀerential op erator o n X S . By deﬁnition S m ( ψ ) 2 = P a k X a ψ k 2 L 2 where the sum is ov er all the k -tuples a with 0 ≤ k ≤ m and where X ∅ ψ stands for ψ . Prop osition 10.7. Ther e e x ist p 2 > 0 such that, for al l suﬃciently smal l ε , one c an cho ose - a non-n e gative smo oth f unc tion ρ ε on W wi th supp ort in W ε such that R W ρ ε = 1 , - a non - n e gative smo oth function τ ε on Y S with s upp ort in Y ε such that τ ε ≤ 1 on Y S and τ ε | Y 4 ε = 1 . - Mor e over, let ϕ ε b e the function on X S deﬁne d b y ϕ ε ( x ) = X { ( y,w ) ∈ Y ε × W ε | y w = x } τ ε ( y ) ρ ε ( w ) . The c h oic es c an b e m a de so that ϕ ε is U ′ f -invariant and S m ( ϕ ε ) ≤ ε − p 2 . W e remark that the sum deﬁning ϕ ε is a ﬁnite sum and hence ϕ ε is w ell deﬁned. T o prov e Prop osition 10.7, we ﬁrst need a lemma whic h constructs some test functions α ε near e . 28 YVES BENOIST AND HEE OH Lemma 10.8. F or a given m ≥ 0 , ther e exists p ∈ N , such that, for al l suﬃciently smal l ε > 0 , one c an cho ose smo oth non-ne gative functions β ε on H ε and smo oth non-ne gative functions ρ ε on W ε satisfying the fol low ing: - one has β ε ≥ 1 on H ε 2 . - one has R W ρ ε dµ W = 1 - if α ε denotes the smo oth function on H ε W ε given by α ε ( hw ) = β ε ( h ) ρ ε ( w ) , then α ε is U ′ f -invariant and S m ( α ε ) ≤ ε − p . Pr o of. The general case reduces to the case of S = S ∞ , b y considering tensor pro ducts with ch aracteristic functions of U ε 0 ∩ H a nd of U ε 0 ∩ W . Hence, w e can assume that S = S ∞ so that G S is a real Lie g roup. Set d = dim W . Fix some smo oth non- negativ e functions β on h := ⊕ v ∈ S h v and ρ on s with supp o rt in a suﬃcien tly small neigh b orho o d of 0 suc h that β (0) > 1 and R s ρ = 1. Then, for suitable constan t s c ε > 0 con v erging to 1, the functions g iv en by β ε ( exp ( X )) = β ( X/ ε ) and ρ ε ( exp ( Y )) = c ε ε − d ρ ( Y /ε ) , for X (resp. Y ) in a ﬁxed compact neighborho o d o f 0 in h (resp. s ), satisfy the prop erties listed ab ov e.  Pr o of of Pr op osition 10 . 7. W e choose the function ρ ε from Lemma 10.8. T o construct the function τ ε , consider a maximal fa mily G ε of p oin ts y ∈ Y ε suc h tha t the subsets y H ε 3 of Y S are disjoin t a nd meet Y 2 ε and let F ε ⊂ G ε the subfamily for whic h y H ε 3 meets Y 4 ε . F or all y ∈ G ε the volume s µ Y S ( y H ε 3 ) are equal and of order ε 3 d with d = dim( H ∞ ). Since µ Y S ( Y S ) = 1 , the cardinalit y of G ε is at most O ( ε − 3 d ). F or y ∈ G ε w e deﬁne a test function β y ,ε on Y S with supp ort on y H ε b y β y ,ε ( y h ) = β ε ( h ) and let β G , ε := P y ∈G ε β y ,ε . Since B ε 3 B ε 3 ⊂ B ε 2 , the sets y H ε 2 , y ∈ G ε , co ver Y 2 ε . Hence β G , ε ≥ 1 on Y 2 ε . F or each y ∈ F ε , consider the t est function τ y ,ε on Y S with supp ort in Y 2 ε giv en b y τ y ,ε := β y ,ε /β G , ε on Y 2 ε and set τ ε := P y ∈F ε τ y ,ε . Note that 0 ≤ τ ε ≤ 1 on Y S , τ ε | Y 4 ε = 1 and τ ε | Y S − Y ε = 0. F or y ∈ F ε , w e also deﬁne the test function ϕ y ,ε on X S with supp ort on y H ε W ε giv en by ϕ y ,ε ( y hw ) := τ y ,ε ( y h ) ρ ε ( w ) = α ε ( hw ) /β G , ε ( y h ) . These f unctions ϕ y ,ε are well-deﬁn ed since y b elongs to the set Y ε giv en by Lemma 10.6. By construction, w e ha ve ϕ ε = P y ∈F ε ϕ y ,ε . It follow s from S m ( α ε ) ≤ ε − p that there exists p 0 > 0 suc h that max y ∈F ε S m ( ϕ y ,ε ) = O ( ε − p 0 ) S - INTEGRAL POINTS 29 and hence S m ( ϕ ε ) = O ( ε − 3 d − p 0 ).  11. Effective equidistribution of transla tes of H S -orbits The goal of this section is to pro v e Theorem 1.9, o r its stronger ve rsion Theorem 11.5 b elo w. This is a n eﬀectiv e v ersion of Prop osition 4.1 on t he equidistribution of translates of H S -orbits in X S . Deﬁnition 11.1. We say that the tr anslates Y S g ar e eﬀe ctively e quidistribute d in X S as g → ∞ i n Z S if ther e exists m ∈ N and r > 0 such that, for any c omp act op en sub gr oup U f of G f and any c omp act subset C of X S , ther e exists c = c ( U f , C ) > 0 satisfying that for any sm o oth function ψ ∈ C ∞ c ( X S ) U f with supp ort in C , one has fo r al l g ∈ G S | Z Y S ψ ( y g ) d µ Y S ( y ) − Z X S ψ dµ X S | ≤ c S m ( ψ ) H S ( z 0 g ) − r . (11.2) Assume further that H is a symmetric K -subgroup o f G . T aking the pro duct of the p olar decomp ositions G v = H v A v K v giv en in Lemma 3.3 ov er v ∈ S , w e obtain a p olar decomp osition of the shap e G S = H S A S K S and w e set F S := A S K S . The followin g eﬀectiv e v ersion of the w av efron t prop erty 3.1 is a main tech nical reason wh y o ur pro of of Theorem 11.5 w or ks for H a symmetric subgro up. Lemma 11.3. Ther e exi sts l ∈ N such that for al l smal l ε, ε ′ > 0 and al l g ∈ F S , (11.4) H S B ε/l U ε ′ /l g ⊂ H S g B ε U ε ′ . Pr o of. W e only hav e to c heck this separately at eac h place v . This statement is then a strengthening of Prop osition 3 .2 on the w av efron t prop erty and is an output of the pro of of this Prop osition.  Theorem 11.5. If H is a symmetric K -sub gr oup of G , then the tr anslates Y S g ar e eﬀe ctively e quidistribute d in X S as g → ∞ in Z S . Pr o of. Since G S = H S F S , it suﬃces to prov e the ab o ve claim fo r g ∈ F S . W e ma y also a ssume that R X S ψ dµ X S = 1. W e w ant to b ound | I g − 1 | where I g := Z Y S ψ ( y g ) dµ Y S ( y ) . W e follo w the pro o f o f Prop osition 4.1. The main mo diﬁcation will b e to replace the c haracteristic function 1 Y ε W b y the test f unction ϕ ε constructed in Prop osition 10.7. By the same argumen t as in section 4, but using the stronger version 11 .3 of the w av efron t lemma, w e ha v e tha t for all small ε > 0 and for any w ∈ W ε (11.6) | I g − Z Y S ψ ( y w g ) d µ Y S ( y ) | ≤ l ε C ψ . 30 YVES BENOIST AND HEE OH Here C ψ is the Lipsc hitz constan t at ∞ , i.e. the smallest constan t suc h that fo r all ε > 0, | ψ ( xu ) − ψ ( x ) | ≤ C ψ ε for all x ∈ X S and u ∈ B ε . Set τ ε , ρ ε and ϕ ε the functions constructed in Prop osition 10 .7 a nd J g ,ε := Z W ε Z Y S ψ ( y w g ) ρ ε ( w ) dµ Y S ( y ) dµ W ( w ) . By in tegrating (11.6) against ρ ε , w e obtain | I g − J g ,ε | ≤ l ε C ψ . Set also K g ,ε := Z X S ψ ( xg ) ϕ ε ( x ) dµ X S ( x ) = Z W ε Z Y S ψ ( y w g ) τ ε ( y ) ρ ε ( w ) dµ Y S ( y ) dµ W ( w ) . Noting that τ ε ( y ) = 1 for y ∈ Y 4 ε , we hav e for some c 1 , p 1 > 0 | J g ,ε − K g ,ε | =     Z W ε Z Y S ψ ( y w g )(1 − τ ε ( y )) ρ ε ( w ) dµ Y S ( y ) dµ W ( w )     ≤ 2 µ Y S ( Y S − Y 4 ε )( Z W ε ρ ε ) k ψ k ∞ ≤ c 1 ε p 1 k ψ k ∞ . Note that K g ,ε = h g .ψ , ϕ ε i . Since ϕ ε and ψ are U ′ f -in v ariant, b y Theorem 9.2 a nd Prop osition 10.7 , w e deduce for some c ′ , c 2 , p 2 > 0 | K g ,ε − Z X S ϕ ε dµ X S | ≤ c ′ kD m ( ψ ) k L 2 kD m ( ϕ ε ) k L 2 H S ( z 0 g ) − κ ≤ c 2 ε − p 2 kD m ( ψ ) k L 2 H S ( z 0 g ) − κ . Moreo ve r, one has     Z X S ϕ ε dµ X S − 1     =     Z Y S τ ε ( y ) dµ Y S ( y ) − 1     ≤ µ Y S ( Y − Y 4 ε ) ≤ c 1 ε p 1 . Since C is compact, the C 1 -norm of a U ′ f -in v ariant f unction ψ supp ort ed on C is b ounded ab o ve b y a unifor m m ult iple of a suitable Sob olev norm as in [2, Theorem 2.20] i.e., one has an inequalit y max( k ψ k ∞ , C ψ ) ≤ c ′′ S 2 m ( ψ ) S - INTEGRAL POINTS 31 with c ′′ = c ′′ ( U f , C ) > 0 indep enden t of ψ . Hence, putting all these upp er b ounds together and using the inequalit y 1 ≤ k ψ k ∞ , w e get | I g − 1 | ≤ | I g − J g ,ε | + | J g ,ε − K g ,ε | + | K g ,ε − R ϕ ε | + | R ϕ ε − 1 | ≤ l ε C ψ + c 1 ε p 1 k ψ k ∞ + c 2 ε − p 2 kD m ( ψ ) k L 2 H S ( z 0 g ) − κ + c 1 ε p 1 ≤ ( c ′ 1 ε p 1 + c ′ 2 ε − p 2 H S ( z 0 g ) − κ ) S 2 m ( ψ ) . Note in the ab o ve that the p ositive constan ts c ′ i , p i , i = 1 , 2, a re indep endent of ψ . No w by taking ε = H S ( z 0 g ) − r /p 1 with r = κ p 1 p 1 + p 2 , we o bt a in as required | I g − 1 | ≤ c S 2 m ( ψ ) H S ( z 0 g ) − r . This concludes the pro of.  Remarks (1) One could also, a s an output of our pro of , compute explicitly m and r and describ e ho w the constant c dep ends on the compact sets U f and C . (2) Note that the ab ov e theorem 11.5 is precisely the eﬀective v ersion of Prop osition 4.1 , since we hav e sho wn that the eﬀectiv e mixing theorem 9.2 together with the eﬀectiv e w av e front lemma 1 1.3 imply the eﬀectiv e equidistribution of Y S g . 12. Effective counting and equidistribution The follo wing deﬁnition is an eﬀectiv e v ersion of Deﬁnition 6.1. Rec all that B ε = B ( e, ε ) is the ball of cen ter e and radius ε in G ∞ (10.4) and that H S is a heigh t function o n Z S as deﬁned in (9.1). Deﬁnition 12.1. A se quenc e of subse ts B n in Z S is said to b e eﬀe ctively wel l- r ounde d if (1) it is invariant under a c omp act op en sub gr oup of G f , (2) ther e exists κ > 0 such that, uniformly for al l n ≥ 1 and al l ε ∈ ]0 , 1 [ , µ Z S ( B + n,ε − B − n,ε ) = O ( ε κ µ Z S ( B n )) wher e B + n,ε = B n B ε and B − n,ε = ∩ u ∈ B ε B n u , (3) for any k > 0 , ther e exists δ > 0 such that, uniformly for a l l n ≫ 1 and al l ε ∈ ]0 , 1[ , one has Z B + n,ε H − k S ( z ) d µ Z S ( z ) = O ( µ Z S ( B n ) 1 − δ ) . If S ∞ is empt y , then the a ssumption (2) is v oid. A subset Ω o f Z S is said to b e eﬀe ctively w el l-r ounde d if the constan t sequence B n = Ω is eﬀectiv ely w ell-ro unded. This means that Ω is of non-empty interior 32 YVES BENOIST AND HEE OH and that the v olume µ Z S ( ∂ ε Ω) of the ε -neigh b orho o d of the b oundary of Ω is a O ( ε κ ) for ε small. F or instance, a compact subset of Z S ∞ with piecewise smo ot h (or ev en piecewise C 1 ) b oundary is eﬀectiv ely w ell-rounded in Z S ∞ . Theorem 12.2. Supp ose that the tr anslates Y S g b e c ome eﬀe ctively e quidistribute d in X S as g → ∞ in Z S . Then for any eﬀe ctively wel l-r ounde d se quenc e of subsets B n in Z S such that vol( B n ) → ∞ ther e exists a c onstant δ 0 > 0 such that # z 0 Γ S ∩ B n = v ol( B n )(1 + O (v ol( B n ) − δ 0 )) . Pr o of. Set Γ S,H := Γ S ∩ H S . As in sections 5 and 6, w e deﬁne a function F n on X S = Γ S \ G S b y F n ( x 0 g ) = X γ ∈ Γ S,H \ Γ S 1 B n ( z 0 γ g ) , for g ∈ G S . F or instance, one has F n ( x 0 ) = # z 0 Γ S ∩ B n . Let m and r b e the in tegers giv en b y Theorem 11.5 and U f a compact op en subgroup of G f . By Lemma 10.8, there exists p > 0, a smo oth U f -in v ariant function α ε on G S , supp orted on B ε U f suc h tha t R G S α ε = 1 a nd S m ( α ε ) ≤ ε − p . Here w e tak e ε and U f small enough so that B ε U f injects to X S , and hence w e ma y consider α ε as a function o n X S . W e also in tro duce the functions F ± n on X S : F ± n,ε ( x 0 g ) = X γ ∈ Γ S,H \ Γ S 1 B ± n,ε ( z 0 γ g ) , for g ∈ G S . Then F − n,ε ( x 0 g ) ≤ F n ( x 0 ) ≤ F + n,ε ( x 0 g ) for all g ∈ B ε × U f and hence h F − n,ε , α ε i ≤ F n ( x 0 ) ≤ h F + n,ε , α ε i . Note that h F ± n,ε , α ε i = Z B ± n,ε  Z Y S α ε ( y g ) d µ Y S ( y )  dµ Z S ( z 0 g ) . Set v n := v o l( B n ) and v ± n,ε := v ol ( B ± n,ε ). Then b y Theorem 11.5 and the assumptions (2 ) a nd (3) o f the deﬁnition 12.1, there exist p o sitiv e constan ts κ , δ and c i suc h tha t for a ll n ≫ 1 and small ε > 0, |h F ± n,ε , α ε i − v n | ≤ Z B + n,ε     Z Y S ( α ε ( y g ) − 1) dµ Y S ( y )     dµ Z S ( z 0 g ) + ( v + n,ε − v − n,ε ) ≤ c S m ( α ε ) Z B + n,ε H S ( z ) − k dµ Z S ( z ) + c 2 ε κ v n ≤ c 1 ε − p v 1 − δ n + c 2 ε κ v n . S - INTEGRAL POINTS 33 Setting δ 0 := δ 1+ p/κ and c ho osing ε = v − δ 0 /κ n w e get F n ( x 0 ) = v n (1 + O ( v − δ 0 n )).  Corollary 12.3. L et S = S 0 ⊔ S 1 b e a p artition of S . Ther e exist δ 0 , c > 0 such that for any eﬀe ctively wel l-r ounde d se quenc e of subsets B n in Z S 1 whose volumes v n := µ Z S 1 ( B n ) tend to ∞ an d for any c o mp act eﬀe ctively wel l-r ounde d subset Ω of Z S 0 , we have # z 0 Γ S ∩ (Ω × B n ) = µ Y S ( Y S ) µ X S ( X S ) v n µ Z S 0 (Ω) (1 + O ( v − δ 0 n )) . Pr o of. T o apply Theorem 12.2, w e only hav e to c hec k that t he sequence A n := Ω × B n of subsets of Z S is eﬀectiv ely well-rounded, which is straig h tforward.  13. Effective well-roundedness In this section, w e g iv e explicit examples of eﬀectiv ely we ll-rounded f a milies (see Deﬁnition 12.1). W e keep the notations for K , S , G , H , Z , H S ( z ) = Q v ∈ S k z k v etc., from the b eginning of section 9. W e also set fo r T > 0, (13.1) B v ( T ) := { z ∈ Z v | k z k v ≤ T } and S v ( T ) := { z ∈ Z v | k z k v = T } . Prop osition 13.2. (1) Fix a subset S 0 ⊂ S c ontaining S ∞ . F or an | S | -tuple m = ( m v ) of p os itive numb ers, deﬁne Z ( m ) := Y v ∈ S 0 B v ( m v ) × Y v ∈ S \ S 0 S v ( m v ) . Then the family of sets Z ( m ) , m v ≫ 1 , pr ovide d non-em pty, is eﬀ e ctively wel l-r ounde d. (2) The family of height b al ls B T = { z ∈ Z S : H S ( z ) ≤ T } is eﬀe ctively wel l-r ounde d. Pr o of. The pro o f relies hea vily on the app endix 16. F or (1), we ma y assume that S contains only one place v . When v is inﬁnite, the condition 12.1 (2) is Lemma 7.8 (2) and 12.1 (3) is Coro llary 16.3.c. When v is ﬁnite, the condition 12.1 (2) is empt y a nd the condition 1 2.1 (3) is Corollary 16.7.b. F or (2), 12.1 (2) is Lemma 7 .8 (4), and 12.1 (3) is a com bination of t he f ollo wing lemma 1 3.3 with the facts tha t, on o ne hand one has B T ,ε ⊂ B k T for some ﬁxed k > 0 and, on the other hand, o ne has V k T = O ( V T ) again b y Lemma 7 .8 (4).  Lemma 13.3. L et B T = { z ∈ Z S : H S ( z ) ≤ T } and V T := µ Z S ( B T ) . Then, for any k > 0 , ther e exists δ > 0 such that Z B T H S ( z ) − k dµ Z S ( z ) = O ( V 1 − δ T ) . 34 YVES BENOIST AND HEE OH Pr o of. W e ma y assume that, fo r all v in S and z in Z v , one has k z k v ≥ 1 . Set b T = { z ∈ Z S ∞ : H S ∞ ( z ) ≤ T } and v T = µ Z S ∞ ( b T ). W e ﬁrst claim that there exists δ > 0 and C > 0 suc h that for an y T > 0, Z b T H − k S ∞ dµ Z S ∞ < C v 1 − δ T . (13.4) Set u T to b e the left hand side of the ab ov e inequality . F o r T large, by Lemma 7.8 (1) one has v T = O ( T m 0 ) f o r some m 0 > 0, hence the deriv ativ e u ′ T = T − k v ′ T satisﬁes u ′ T = O ( v − k /m 0 T v ′ T ) and, in tegrating, one gets u T = O ( v 1 − k /m 0 T ). F or T b ounded, since H S ∞ is b ounded b elow, o ne gets u T = O ( v T ). Putting this together, one gets (13.4). No w, for any t uple m = ( m v ) ∈ N S f , set S ( m ) := Q S v ( m v ) ⊂ Z S f where S v ( m v ) := { z ∈ Z v : k z k v = m v } . Also set π m := Q v m v ∈ N and ω m = µ Z S f ( S ( m )). Then w e hav e, where the following sums are tak en ov er the tuples m ∈ N S f for whic h S ( m ) is non empt y , Z B T H S ( z ) − k dµ Z S ( z ) = P m  R b T /π m H − k S ∞ dµ Z S ∞   R S ( m ) H − k S f dµ Z S f  = P m π − k 2 m  R b T /π m H − k S ∞ dµ Z S ∞   R S ( m ) H − k 2 S f dµ Z S f  ≤ C P m π − k 2 m  v T /π m ω m  1 − δ ≤ C  P m π − k 2 δ m  δ  P m v T /π m ω m  1 − δ ≤ C  Q v ∈ S f (1 − q − k 2 δ v )  − δ V 1 − δ T , with p ositiv e constan ts C and δ giv en by (13.4) and Corollary 16 .7.b.  14. Effective applica tions In this section, assuming K is a num b er ﬁeld and H is a symmetric K -subgroup, w e giv e pro ofs of eﬀectiv e v ersions of our main theorems listed in the in tro duction, k eeping the not ations therein. Corollary 14.1. Assume that Z O S 6 = ∅ . Th en for any ﬁnite v ∈ S , ther e exists δ > 0 such that f o r any eﬀe ctively w el l-r ounde d subset Ω ⊂ Z S −{ v } , # { z ∈ Z O S ∩ Ω : k z k v = T } = f v ol (Ω × S v ( T )) (1 + O ( T − δ )) as T → ∞ subje ct to S v ( T ) 6 = ∅ . Recall that the normalized volume f v ol has b een deﬁned in (1.5). This corollary is an equidistribution statemen t since one has f v ol (Ω × S v ( T )) = C µ Z S −{ v } (Ω) µ Z v ( S v ( T )) S - INTEGRAL POINTS 35 with a constant C indep enden t of n and Ω. Pr o of of Cor ol lary 14.1 : The same claim with the error term O ( T − δ ) replaced b y O (( µ Z v ( S v ( T )) − δ ) fo llows immediately from Theorems 11.5, 12.2 and Prop osition 13.2. Now b y Corollary 16.7.c, w e hav e a constan t a > 0 suc h that for T la r ge and S v ( T ) 6 = ∅ , µ Z v 0 ( S v ( T )) − 1 = O ( T − a/ 2 ) . This pro v es the claim.  F or z ∈ V Q , the condition k z k p ≤ p n is equiv alen t to z ∈ p − n V Z . Hence we obtain: Corollary 14.2. Assume K = Q a nd ﬁx a prime p such that Z p is non c omp act and Z Z [ p − 1 ] is non emp ty. T h en ther e exists δ > 0 such that, for any no n -empty c omp act subset Ω ⊂ Z R with pie c ewise smo oth b oundary, # Ω ∩ p − n V Z = f v ol (Ω × B p ( p n )) (1 + O ( p − δn )) as n → ∞ , wher e B p ( p n ) is deﬁne d in ( 13.1) . Note ag ain that f v ol (Ω × B p ( p n )) = C µ Z R (Ω) µ Z p ( B p ( p n )) for some C > 0 inde- p enden t of n a nd Ω. Pr o of of The or ems 1.3 and 1.7 Letting v T := v ol( B S ( T )), Theorem 1.3 with O ( T − δ ) r eplaced by O ( v − δ T ) immediately follo ws f r o m Theorems 11 .5 and 12.2 and Prop osition 13.2. T o obtain the giv en error term, note that at least one of the factors Z v 0 is non-compact. Fix R 0 > 1 suc h that the v o lume of B S \{ v 0 } ( R 0 ) = { z ∈ Z S − v 0 : H S \{ v 0 } ( z ) ≤ R 0 } is p o sitive. Since B S ( T ) con ta ins the pro d- uct B v 0 ( T R − 1 0 ) B S \{ v 0 } ( R 0 ), w e ha v e v − 1 T = O ( µ Z v 0 ( B v 0 ( T R − 1 0 )) − 1 ) . By Corollary 16.3.d for v 0 arc himedean and Corollary 16.7.c fo r v 0 non-arc himedean, w e ha v e a constan t a > 0 satisfying µ Z v 0 ( B v 0 ( T )) − 1 = O ( T − a/ 2 ) . This prov es the claim. The same pro of works fo r Theorem 1.7 applying Corollary 12.3 in place of Theorem 12.2.  Pr o of of Cor ol lary 1.10. All three cases ﬁt in our setting as in the introduction. F or (A), if f has signature ( r, s ), V 1 ( R ) can b e iden tiﬁed with Spin( r − 1 , s ) \ Spin( r , s ) where the Spin( r , s )-action on R r + s is giv en through the pro- jection Spin( r, s ) → SO( r, s ). F or (B): w e hav e the a ction o f G = SL n on V b y ( g , v ) 7→ g t v g . And V 1 ( R ) is a ﬁnite disjoin t union of SO( r, s ) \ SL n ( R ) fo r r + s = n , eac h of them b eing the v ariet y consisting of symmetric mat r ices of signature ( r, s ). F or (C), we hav e V 1 ( R ) = Sp 2 n ( R ) \ SL 2 n ( R ) with the action ( g , v ) 7→ g t v g . 36 YVES BENOIST AND HEE OH Note for (A), if n = 3, H R = Spin(1 , 1) ma y arise and the additional assumption that f do es not represen t 0 ov er Q implies that H do es not allow any non-tr ivial Q -c haracter. In all other cases, H is semisimple a nd hence has no c haracter. No w w e give a uniform pro of assuming that S = {∞ , p } for the sak e of sim- plicit y . It is easy to generalize the argumen t for a general S . Also note that this pro of w orks equally w ell for an y homog eneous integral p olynomial f whose lev el set can b e iden tiﬁed with a symmetric v ariet y in our set-up. Let d = deg f . F or eac h 0 ≤ j ≤ d − 1, consider the radial pro jection π j : V p kd + j → V p j giv en b y x 7→ p − k x . Then since the degree of f is d a nd the radial pro jection is bijectiv e, # Ω ∩ π ( V p kd + j ( Z ) prim ) = # { z ∈ V p j ( Z [ p − 1 ]) ∩ Ω j : k z k p = p k } where Ω j := p j /d Ω ⊂ V p j . Since V p j ( Z [ p − 1 ]) is a ﬁnite union of G ( Z [ p − 1 ])-orbits, w e o btain by Corollary 1 2.3 with S 0 = {∞ } a nd S 1 = { p } # Ω ∩ π ( V p kd + j ( Z ) prim ) ∼ ω p kd + j v ol (Ω)(1 + O ( ω − δ p kd + j )) where ω p kd + j = µ ( { x ∈ V p j ( Q p ) : k x k p = p k } ). Note that, by R emark 8.3, ω m go to inﬁnit y with m when it is non zero.  Pr o of of Cor ol lary 1.11 . As b efore, w e a ssume S = {∞ , p } for simplicit y . W e use the same notatio n as in the ab o ve pro of. Then for eac h ﬁxed 0 ≤ j ≤ d − 1, f ( x ) = p k d + j is equiv alent to f ( p − k x ) = p j and, if z = p − k x with x ∈ V ( Z ) prim , one has k z k ∞ k z k p = k x k ∞ . Therefore N j,T := # { x ∈ V ( Z ) prim : k x k ∞ < T , f ( x ) = p k d + j for some in teger k ≥ 0 } = # { z ∈ V p j ( Z [ p − 1 ]) : k z k ∞ k z k p < T } . By Theorem 1.3, one has N j,T = v j,T (1 + O ( v − δ j j,T )) where v j,T = f v ol ( { ( z ∞ , z p ) ∈ V p j ( R ) × V p j ( Q p ) : k z ∞ k ∞ k z p k p < T } ) . Since # { x ∈ V ( Z ) prim : k x k ∞ < T , f ( x ) ∈ p Z } = P j N j,T , and v T = P j v j,T , this pro ves the claim.  15. More examples Here are a few concrete examples of applications of Theorem 1 .4 to emphasize the meaning of our results. F or eac h of them, w e ha ve sele cted a sp eciﬁc global ﬁeld K with sets S 0 , S 1 (most of ten K = Q , S 0 = { v 0 } a nd S 1 = { v 1 } ) and w e hav e selected a classical symmetric space Z deﬁned ov er K . W e lo ok at the repartition of S -integral p oin ts z in Z v 0 when imp osing conditions o n the v 1 -norm of z . S - INTEGRAL POINTS 37 Symmetric matrices with t wo real places. This example is v ery classical. Let τ b e the non trivial automor phism of the real quadra t ic ﬁeld K := Q [ √ 2] and set, for d ≥ 2, Z {∞} := { M ∈ M d ( R ) p ositiv e deﬁnite symmetric matrix of determinan t 1 } and Z n := { M ∈ Z {∞} ∩ M d ( Z [ √ 2]) | P i,j τ ( m i,j ) 2 ≤ n 2 } . Lemma 15.1. As n → ∞ , these discr ete sets Z n b e c o me e quidistribute d in the non-c omp act Riemannian symmetric sp ac e Z {∞} . Pr o of. Let v 0 and v 1 b e the tw o inﬁnite places o f K : f o r λ ∈ K , | λ | v 0 = | λ | and | λ | v 1 = | τ ( λ ) | . Apply Theorem 1.4 to K = Q [ √ 2] , S 0 = { v 0 } , S 1 = { v 1 } , Z = SL d / SO d , (15.1) and to the group G = SL d whic h acts by M → g M t g on the v ector space V of symmetric d × d -matrices, with Z ∼ { M ∈ V | det( M ) = 1 } as a G -o r bit. Note tha t the g roup SL( d , R ) acts transitiv ely on Z {∞} .  Orthogonal pro jections with one real and one ﬁnite place. This example is also quite classical. Let p b e a prime num b er, d = d 1 + d 2 ≥ 3, Z {∞} := { π ∈ M d ( R ) π 2 = t π = π and tr( π ) = d 1 } the Grassmannian of R d , a nd Z n := { π ∈ Z {∞} | p n π ∈ M d ( Z ) } . Lemma 15.2. As n → ∞ , these discr ete sets Z n b e c o me e quidistribute d in the c omp act Riemannian symmetric sp ac e Z {∞} . Pr o of. Apply Theorem 1 .4 and Remark 8.3 with K = Q , S 0 = {∞ } , S 1 = { p } , Z = O d / O d 1 × O d 2 , (15.2) and to the group G = Spin d whic h acts, b y conjuga tion via SO d , on the v ector space V of d × d - matrices, with Z ∼ { π ∈ V | π 2 = t π = π and tr( π ) = d 1 } as a G -orbit. Note tha t the g roup Spin ( d, R ) acts transitive ly on Z {∞} .  Complex struct ur es wit h one ﬁnite and one real place. In this example, one c ho oses a prime n um b er p a nd set, for d ≥ 1, Z { p } := { J ∈ M 2 d ( Q p ) | J 2 = − Id and tr ( J ) = 0 } and Z R := { J ∈ Z { p } ∩ M 2 d ( Z [ 1 p ]) | P i,j J 2 i,j ≤ R 2 } . Lemma 15.3. As R → ∞ , these discr ete sets Z R b e c o me e quidistribute d in the p -adic symmetric s p ac e Z { p } . 38 YVES BENOIST AND HEE OH Pr o of. Apply Theorem 1 .4 to K = Q , S 0 = { p } , S 1 = {∞ } , Z = GL 2 d / GL d × GL d , (15.3) and to the group G = SL 2 d whic h acts b y conjugation on the vec tor space V o f 2 d × 2 d -matrices, with Z ∼ { J ∈ V | J 2 = − Id and tr ( J ) = 0 } a s a G -o rbit. Note tha t the g roup SL(2 d, Q p ) acts transitiv ely on Z { p } .  An tisymmetric matrices with tw o ﬁnite places in c haracteristic zero. In this example, one c ho oses t w o distinct prime n umbers p and ℓ , and set, for d ≥ 1, n ≥ 0 , and R > d , Z { p } := { A ∈ M 2 d ( Q p ) | A = − t A and det( A ) = 1 } and Z n,R := { A ∈ Z { p } | P i,j A 2 i,j < R and ℓ n A ∈ M 2 d ( Z [ 1 p ]) } . Lemma 15.4. As n + R → ∞ , these discr ete sets Z n,R b e c o me e quidistribute d in the p -adic symme tric sp ac e Z { p } . Pr o of. Apply Theorem 1 .4 and Remark 8.3 with K = Q , S 0 = { p } , S 1 = {∞ , ℓ } , Z = SL 2 d / Sp d , (15.4) and to the group G = SL 2 d whic h acts by g → g A t g on the v ector space V of an tisymmetric 2 d × 2 d -matrices, with Z ∼ { A ∈ V | det( A ) = 1 } as a G -or bit. Note tha t the g roup SL 2 d ( Q p ) acts transitiv ely on Z { p } .  Quadrics with t w o places in p ositive ch aracteristic. In this example, p is an o dd prime, and one set, for d ≥ 3, Z { 0 } := { P ∈ F p (( t )) d | P 2 1 + · · · + P 2 d = 1 } and, for n 1 ≥ n 2 ≥ . . . ≥ n d ≥ 0, Z n 1 ,...,n d := { P ∈ Z { 0 } ∩ F p [ t, t − 1 ] d | deg( P i ) = n i ∀ i } , where deg( P a i t i ) := max { i | a i 6 = 0 } . Lemma 15.5. If n 1 = n 2 = n 3 go es to inﬁnity or if p ≡ 1 mo d 4 and n 1 = n 2 go es to inﬁnity, these dis cr ete sets Z n b e c o me e quidistribute d in the spher e Z { 0 } . Pr o of. Let 0 and ∞ b e the t wo (ﬁnite) places of the ﬁeld F p ( t ) a sso ciat ed t o the t wo p oints 0 a nd ∞ of P 1 ( F p ). Apply Theorem 1.4 and Remark 8.3 with K = F p ( t ) , S 0 = { 0 } , S 1 = {∞ } , Z = SO d / SO d − 1 , (15.5) and to t he group G = Spin d whic h acts naturally , via SO d , on the d - dimensional v ector space V , with the sphere Z ∼ { v ∈ V | v 2 1 + . . . v 2 d = 1 } as a G - orbit. The corresp onding t wo completions are the ﬁelds of La uren t series K 0 = F p (( t )) and K ∞ = F p (( t − 1 )), and the ring of S -integers is O S = F p [ t, t − 1 ]. Set Z {∞} := { P ∈ F p (( t − 1 )) d | P 2 1 + · · · + P 2 d = 1 } , S - INTEGRAL POINTS 39 note that the w ell-rounded subset B n 1 ,...,n d := { P ∈ Z {∞} | deg( P i ) = n i ∀ i } is non-empt y if and o nly if n 1 = n 2 = n 3 or p ≡ 1 mo d 4 and n 1 = n 2 . Note also that the gro up Spin( d, F p (( t ))) acts transitiv ely on Z { 0 } .  Other examples. The reader may construct easily many similar examples c ho o sing o t her t r iples ( K, S, Z ). F or instance, “Quadrics with three inﬁnite places”, “La g rangian decomp ositions with t wo inﬁnite and three ﬁnite place”, “Hermitian matrices with four places in p ositiv e characteristic”, and so o n.... 16. Appe ndix: V olume of b alls In this app endix w e prov e precise estimates for the v olume of balls whic h are needed in sections 7 and 13. These estimates will b e consequences of the follo wing t wo general theorems 1 6.1 and 16.6. V olume of balls o v er the reals. W e will ﬁrst need a v ariation of a theorem on ﬁb er integration. This theorem sa ys that the volume of the ﬁb ers of an analytic function has a, term-by -term diﬀeren tiable asymptotic expansion in the scale of functions t j (log t ) k with j rational and k non-negative integer. More precisely , Theorem 16.1. [30] L et X ⊂ R m b e a smo oth r e al ana l ytic va riety, f : X → R a r e al analytic function and ν a C ∞ me asur e on X . Then, for any c omp act K of X , ther e exist d ∈ N and a set { A j,k : j ∈ 1 d N , k ∈ Z , 0 ≤ k < m } of distributions on X s upp orte d by f − 1 (0) s uch that, for eve ry C ∞ function ϕ : X → R with supp ort in K , the inte g r al v ϕ ( t ) := Z 0 ≤ f ( x ) ≤ t ϕ ( x ) dν ( x ) has a term-by-term diﬀer entiable a symptotic exp ansion when t > 0 go es to 0 X j ∈ 1 d N X 0 ≤ k 0 . b) Mor e ov e r d dT v T ∼ c d dT ( T a (log T ) b ) , as T → ∞ . c) F or any k 0 > 0 , ther e exists δ 0 > 0 such that one has , as T → ∞ , Z B T k z k − k 0 dµ ( z ) = O ( v 1 − δ 0 T ) . d) If G is semisimple and Z is non c omp act then one has a 6 = 0 . Remarks - When Z is a symmetric v ariet y , t he p oin t a) is pro ven in [25, Corollary 6.1 0] fo r an y norm on V and the parameters a and b are explicitly giv en. - When G is a g roup of diagonal matrices, the constant a is zero. Pr o of. a) and b) This is a sp ecial case of Prop osition 16.2. Note that since v T is an increasing function of T , one has a ≥ 0 . Moreov er, note that, when a = b = 0, the orbit is of ﬁnite v olume hence compact. c) Set u T := R B T k z k − k 0 dµ ( z ). By a), one has v T = O ( T m 0 ) for some m 0 > 0. Hence the deriv ativ e u ′ T = T − k 0 v ′ T satisﬁes u ′ T = O ( v − k 0 /m 0 T v ′ T ) and, integrating, one gets u T = O ( v 1 − k 0 /m 0 T ). d) This is a sp ecial case of t he f o llo wing Prop osition 1 6 .4  Prop osition 16.4. With the notations of Cor ol lary 16.3, one ha s the e quiva l e nc e: A l l unip otent eleme n ts of G a ct trivia l ly on Z ⇐ ⇒ a = 0 . Pr o of. = ⇒ By assumption the normal subgroup o f G generated by the unip otent elemen ts of G acts trivially on Z . Hence one can assume that G is a pro duct o f a compact g roup by a r -dimensional g r oup o f diagonal matr ices. In t his case o ne has µ ( B T ) = O ((log T ) r ) as T → ∞ . ⇐ = This implication is a conseque nce of the following Lemma 1 6.5.  Lemma 16.5. L et U b e a one-p ar ameter unip otent sub gr oup of GL m ( R ) , µ a U -invariant me asur e on R m which is not supp o rte d by the U -ﬁx e d p oints and set B T for the eucli d e an b al l of r adius T on R m . The n one has lim inf T →∞ log( µ ( B T )) log( T ) > 0 . 42 YVES BENOIST AND HEE OH Pr o of. Fir st of a ll, note that all the or bit s U z of U in R m are images of R b y p olynomial maps t 7→ u t z of degree d z ≤ m . W e may assume that this degree d z is µ - almost eve rywhere non-zero constant. Set d ≥ 1 for this degree, write u t z = t d v z + O ( t d − 1 ) for some non- zero v z ∈ R m , and note that the constan t in volv ed in this O ( t d − 1 ) is uniform on compact subsets of R m . One can ﬁnd a compact subse t C ⊂ R m transv ersal t o the U -actio n suc h t hat µ ( U C ) > 0. The pull-bac k o n R × C of the measure µ b y the actio n ( t, z ) 7→ u t z has the form d t ⊗ ν where dt is the Leb esgue measure o n R and ν is a non-zero measure on C . Cho ose c > sup z ∈ C k v z k . Then, for R large, one has u [0 ,R ] ( C ) ⊂ B cR d and hence µ ( B cR d ) ≥ R ν ( C ). This prov es our claim.  V olume of balls ov er the p -adics. W e will also need Denef ’s theorem on p -adic in tegratio n. F or t ha t w e need some notations. A subset of Q m p is said semialgebr aic if it is obtained b y b o olean o p erations from sets P f ,r := { x ∈ Q m p / ∃ y ∈ Q p : f ( x ) = y r } with f a p olynomial in m v ariables with co eﬃcien ts in Q p and r ≥ 2. According to Macin tyre’s theorem, whic h is the p -adic analog of T arski-Seiden b erg theorem, those sets are exactly the deﬁnable sets of the ﬁeld Q p [33]. A function f b et wee n t w o Q p -v ector spaces is said semialge br aic if its graph is semialgebraic. According t o Denef ’s cell decomp osition theorem ([13] and [9]), for ev ery semialgebraic subs et S , there exists a ﬁnite partition of S in sem ialgebraic sets S 1 , . . . , S j max (called cells) suc h that, for each j = 1 , . . . , j max , S j is in semialgebraic bijection with a semialgebraic op en subset O j of a v ector space Q d j p (recen tly , R. Cluc k ers has shown the existence o f a semialgebraic bijection b et w een S itself and some Q d p ). A measure µ on S is said semialgebr aic if there exists a cell decomp osition o f S on each cell of whic h µ is of the form | g j ( x ) | dx where g j is a semialgebraic function on Q d j p and dx is a Haar measure on Q d j p . A function a : Z → Z is said simple if there are ﬁnite partition of N and − N b y ﬁnite sets and arithmetic progressions on whic h a is aﬃne, see [12, § 2.13, 2.14 and 4.4]. Theorem 16.6. [12, Theorem 3.1] L et µ b e a sem ialgebr aic me asur e on an m - dimensional semialgeb r aic subset S over Q p and f b e a se m ialgebr aic function on S . F or n ∈ Z , s e t I n := Z | f ( x ) | = p n dµ ( x ) when this inte gr al is ﬁn ite and I n = 0 otherwise. Then, for al l n ∈ Z , one has I n = X 1 ≤ i ≤ e γ i ( n ) p β i ( n ) wher e e ∈ N , β i : Z → Z is a simple function and γ i : Z → Z is a pr o duct of at most m sim ple functions for e ach 1 ≤ i ≤ e . S - INTEGRAL POINTS 43 F or instance, an orbit under the group of Q p -p oints of a Q p -algebraic group acting a lg ebraically is deﬁnable and hence semialgebraic, by Macin tyre’s theorem, and an in v ariant measure on this o r bit is semialgebraic. Hence one gets: Corollary 16.7. L et k b e a ﬁn i te extension of Q p , q the absol ute value of an uniformizer, G the g r oup of k -p oints of an algebr aic k -gr oup, ρ : G → GL ( V ) a r epr esentation of G deﬁne d over k , Z a close d G -orbit in V , µ an invariant me asur e on Z and k · k a max norm on V . Denote by S T the spher e S T = { z ∈ Z : k z k = T } and set v T := µ ( S T ) . a) Ther e exists N 0 ∈ N such that, for e ach 0 ≤ j 0 < N 0 , one o f the fol lowing holds: (1) S q j is empty, for j ≡ j 0 mo d N 0 lar ge; (2) ther e exist a j 0 ∈ Q ≥ 0 , b j 0 ∈ Z ≥ 0 , and c j 0 > 0 such that, v q j ∼ c j 0 q a j 0 j j b j 0 for j ≡ j 0 mo d N 0 lar g e. b) F or any k 0 > 0 , ther e exists δ 0 > 0 such that one has , as T → ∞ , Z S T k z k − k 0 dµ ( z ) = O ( v 1 − δ 0 T ) . c) If G is semisimple and Z is n on c omp act then, for al l j 0 in c ase ( 2) , one has a j 0 6 = 0 . Remarks - Let us recall that a max norm is a norm g iv en in some basis e 1 , . . . , e m b y k P x i e i k = max | x i | . - When G is a g roup of diagonal matrices, all the constan ts a j 0 are zero. Pr o of. Viewing V as a Q p v ector space, w e ma y assume that k = Q p . a) This is a sp ecial case of Theorem 16.6. b) By a) , there exists m 0 > 0 suc h that v T = O ( T m 0 ). Hence one ha s R S T k z k − k 0 dµ ( z ) = T − k 0 v T = O ( v 1 − k 0 /m 0 T ) . c) This is a sp ecial case o f the following Prop osition 16.8 whic h is analog ous to Prop osition 16.4.  Prop osition 16 .8. With the notations of Cor ol lary 1 6 .7, the fol lo wing a r e e quiv- alent: ( i ) A l l unip otent el e ments of G act trivia l ly on Z , ( ii ) F or al l j 0 in c ase (2) , one has a j 0 = 0 , ( iii ) Either Z is c omp act or, for some j 0 in c ase (2) , one has a j 0 = 0 . Pr o of. The pro of is as in Prop osition 16.4. ( i ) ⇒ ( ii ) By assumption t he no r mal subgroup o f G generated by the unip oten t elemen ts of G acts trivially on Z . Hence one can assume that G is a pro duct o f a compact group b y an r -dimensional group of diagonal matrices. In this case, one has µ ( S p j ) = O ( j r ) a s j → ∞ . 44 YVES BENOIST AND HEE OH ( ii ) ⇒ ( iii ) If Z is non compact, at least one j 0 is in case (2). ( iii ) ⇒ ( i ) This implication is a consequ ence of the following Lemma 16 .9.  Lemma 16.9. L e t k b e a ﬁni te extens i o n o f Q p , U a one-p ar ameter unip otent sub gr oup of GL( m, k ) , µ a U -invariant m e asur e on k m which is n o t supp orte d by the U -ﬁxe d p oints and denote b y S T the spher e of r adius T on k m for the m ax norm. Then one has, as T → ∞ subje ct to the c ondition µ ( S T ) 6 = 0 , liminf log( µ ( S T )) log( T ) > 0 . Pr o of. The pro of is as in Lemma 1 6.5. First o f all, note that all t he orbits U z of U in k m are images of k b y p olynomial maps t 7→ u t z of degree d z ≤ m . W e ma y assume that t his degree d z is µ -almo st ev erywhere non-zero constan t. Set d ≥ 1 for this degree and write u t z = t d v z + O ( t d − 1 ) for some non-zero v z ∈ k m . Let q b e the absolute v alue of an uniformizer. The set { j ∈ N : U z ∩ S q j 6 = ∅} is then equal, up t o ﬁnite sets, to some arithmetic progr ession j z + N d with 0 ≤ j z < d . W e may assume that this in teger j z is µ -almost eve rywhere constan t. Set j 0 for this in teger. One can ﬁnd a compact subset C ⊂ k m transv ersal to the U -a ction suc h t ha t µ ( U C ) > 0 and on whic h k v z k is constant equal t o some p ow er q j 0 + dm 0 with m 0 ∈ N . The pull-back o n k × C of the measure µ by the action ( t, z ) 7→ u t z has the for m dt ⊗ ν where dt is a Haar measure on k and ν is a non-zero measure on C . 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DMA-ENS 45 rue d’Ulm P aris 7500 5 E-mail addr ess : beno ist@dm a.ens. fr Ma thema tics 25 3-37, Cal tech, P asadena, CA 91 125 current add ress: Ma thema tics, Box 1914, Bro wn University, Pro vidence, RI E-mail addr ess : heeo h@math .brown .edu

Effective equidistribution of S-integral points on symmetric varieties

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