QUANTITATIVE MULTIPLE MIXING MICHAELBJÖRKLUND,MANFREDEINSIEDLER,ANDALEXANDERGORODNIK 7 ABSTRACT. Wedevelopinthispaperageneralandnoveltechniquetoestablishquantitative 1 estimatesforhigherordercorrelationsofgroupactions. Inparticular,weprovethatactionsof 0 semisimpleLiegroups,aswellassemisimpleS-algebraicgroupsandsemisimpleadelegroups, 2 onhomogeneousspacesareexponentiallymixingofallorders.Asacombinatorialapplication n ofourresults,we proveforsemisimpleLiegroupsan effectiveanalogue ofthe Furstenberg- a Katznelson-WeissTheoremonfiniteconfigurationsinsmallneighborhoodsoflattices. J 4 1. INTRODUCTION ] S 1.1. Generalcomments D The aim of this paper is to investigatethe behavior of higher order correlations for group . h actions. Letusconsiderameasure-preservingactionofalocallycompactgroupGonaproba- t a bilitymeasurespace(X,m). Givenatest-functionφ L (X),weobtainafamilyoffunctions m ∈ onXdefinedby [ ∞ g φ : x φ(g−1 x), g G, 1 · 7→ · ∈ v We may think about {g φ : g G} as a collection of random variables on (X,m). For · ∈ 5 "sufficiently chaotic" group actions, it is natural to expect that these random variables are 4 asymptotically independent. The independencepropertyis measuredby the correlations of 9 0 theform 0 m((g φ) (g φ)) = φ(g−1 x) φ(g−1 x)dm(x), 1. 1 · ··· k· ZX 1 · ··· k · 0 whereg ,...,g G. WesaythattheG-actionon(X,m)ismixingoforderkifforeverychoice 7 1 k ∈ offunctionsφ ,...,φ L (X), 1 1 k ∈ : v m((∞g φ ) (g φ )) m(φ ) m(φ ) 1 1 k k 1 k i · ··· · −→ ··· X whenever g−1g in G for every i = j. It is a difficult problem in general to establish r mixingofhiighejro→rders. Forinstance,It6isnotknownwhetherforZ-actionsmixing oforder a two implies mixing ∞of order three, and there are indeed examples of Z2-actions which are mixingofordertwo,butnotmixingoforderthree(see[19]). In thispaperwe developa methodwhich allows us toinductively obtain quantitative es- timatesoncorrelationsoforderkassumingonlyinformation aboutthecorrelationsoforder two. While weare mostlyconcernedwith actions ofsemisimple Liegroupsand semisimple algebraic groups, it will be apparent from the proofs that our method can potentially be ap- pliedinmoregeneralsettingsprovidedthatthereisacollectionofone-parametersubgroups whichsatisfiescertainregularity,growth,andmixingassumptions. The multiple mixing property has been extensively studied for flows on homogeneous spacesoftheformX = Γ\L,whereLisaconnectedLiegroup,andΓ isalatticeinL. Weshall considertheleftactionofLonXdefinedby l x = xl−1 forl Landx X. (1.1) · ∈ ∈ 1 2 MICHAELBJÖRKLUND,MANFREDEINSIEDLER,ANDALEXANDERGORODNIK ItfollowsfromtheworkofDani[7,8]thatundermildassumptions,anypartiallyhyperbolic one-parameter flow on the space X satisfies the Kolmogorov property, so in particular it is mixing of all orders. It was conjectured by Sinai in [37] that the horocycle flow (which is not partially hyperbolic) is also mixing of all orders. Although mixing of order two for the horocycleflowisrelativelyeasytoproveusingrepresentation-theoretictechniques(see[33]), Sinai’sconjecturewasprovedinfullgeneralityonlymuchlaterbyMarcusin[26]. A strikingly general result about mutiple mixing was established by Mozes in [28]. He shows that for arbitrary measure-preserving actions of a connected Lie group G, mixing of ordertwoimpliesmixingofallordersprovidedthatthegroupGisAd-proper(namely,ithas finitecentre,anditsimageundertheadjointmapintothegroupAut(Lie(G))isclosed). This, in particular, applies to connected semisimple Lie group with finite centre. Using Ratner’s measure classification, Starkov in [38] proved mixing of all orders for general mixing one- parameterflowsonfinite-volumehomogeneousspaces. Althoughquantitativeestimatesforcorrelationsofordertwohaveasubstantialhistory,it seems that very little is known regarding effective estimates of correlations of higher order. We intend to remedy this gap in the present paper. Note that the analysis of higher order correlations arises naturally in many combinatorial, arithmetic, and probabilistic problems. In Section 3, we apply our results to prove the existence of "approximate configurations" in latticesinsemisimpleLiegroups. In the forthcoming works[1] and [2], we use our results in this paper to establish quantitative estimates on the number rational points in compactifications of certain homogeneous algebraic varieties, and limit theorems describing the fine statistical properties of subgroup actionsonhomogeneousspaces. 1.2. SemisimpleLiegroups Let G be a connected semisimple Lie group with finite centre. We observe that given a measure-preserving action of G on a probability space (X,m), the correlations of order two can beinterpretedasmatrix coefficientsofthecorrespondingunitaryrepresentationofGon theHilbertspaceL2(X). StartingwiththeresearchprogramofHarish-Chandra(summarized in themonographs[40,41]), propertiesofmatrix coefficientsforrepresentationsofsemisim- pleLiegroupshavebeenextensivelystudied. Inparticular,wementiontheimportantworks of Borel and Wallach [4], Cowling [6], Howe [16], Li and Zhu [20, 21], Moore [27], and Oh [30,31]whichestablishexplicitestimatesonmatrixcoefficientsforsemisimplegroups. Let us formulate the main estimate coming for these works. This estimate will be the startingpointofourinvestigation. Wefixaleft-invariantRiemanniandistanceρ onGwhich G is bi-invariant under a fixed maximal compact subgroup K of G. Let π : G U(H) be a → unitaryrepresentationofG. Wesaythatπhasstrongspectralgapiftherestrictionofπtoevery noncompact simple factor of G is isolated from the trivial representationwith respect to the Fell topology on the dual space. For every unitary representation (H,π) of G with strong spectralgap,thereexistC,δ> 0suchthatforallK-finitevectorsv ,v H, 1 2 ∈ π(g)v ,v 6 Ce−δρG(g,e) N(v )N(v ), (1.2) 1 2 1 2 h i whereN(v) = (dim Kv )1/2 v . Itisimportantforapplicationstohaveananalogueofthees- h i k k timate(1.2)whichisvalidforallsmoothvectorsinH. ItwasobservedbyKatokandSpatzier in [18] that under the strong spectral gap assumption, there exists δ > 0 such that for all sufficientlylargeintegersdandarbitrarysmoothvectorsv ,v H, 1 2 ∈ π(g)v ,v e−δρG(g,e) Cdv Cdv , (1.3) h 1 2i ≪d k G 1kk G 2k QUANTITATIVEMULTIPLEMIXING 3 whereC denotestheCasimirdifferentialoperatorforG. G LetusnowassumethatthegroupGisaclosedsubgroupofa(possiblybigger)connected Lie group L. Let Γ be a lattice subgroup in L and equip X = Γ\L with its Haar probability measure m. We consider the left action of G on X defined by (1.1). We say that this action has strong spectral gap if the corresponding unitary representation of G on L2(X) has strong 0 spectralgap. Ifthisisthecase,thentheestimate(1.3)impliesthatthereexistsδ > 0suchthat forallsufficientlylarged,forallfunctionsφ ,φ C (X),andforallg G,wehave 1 2 c ∈ ∈ |m((g φ )φ )−m(φ )m(φ )| e−δρ∞G(g,e) Cdφ Cdφ . (1.4) · 1 2 1 2 ≪d k G 1k2k G 2k2 Our first main result provides analogous estimates on correlations of arbitrary orders. These estimates will be formulated in terms of certain Sobolev norms S on smooth func- d tionsonX. WeshallpostponetheirdefinitiontoSection2.2below. Theorem1.1 (Exponentialmixing ofall ordersfor Liegroups). LetLbea connectedLiegroup, letΓ bealatticeinL,andletmdenotetheHaarprobabilitymeasureonX = Γ\L. LetGbeaconnected semisimple Lie subgroup of L with finite center, and assume that the action of G on X has strong spectralgap. Then, for every k > 2 and for sufficiently large d, there exists δ = δ(k,d) > 0 such that for all functionsφ ,...,φ C (X)andg ,...,g G,wehave 1 k c 1 k ∈ ∈ |m((g φ ) (g φ∞ ))−m(φ ) m(φ )| M(g ,...,g )−δ S (φ ) S (φ ), 1 1 k k 1 k d,k 1 k d 1 d k · ··· · ··· ≪ ··· where M(g ,...,g ) :=exp minρ (g ,g ) . 1 k G i j i6=j (cid:18) (cid:19) Remark1.2. Theorem1.1shouldbecomparedwiththerecentworkofKonstantoulas[24],in whichestimatesoftheform |m((a φ ) (a φ ))−m(φ ) m(φ )| 6R(a ,...,a ) C(φ ,...,φ ), (1.5) 1 1 k k 1 k 1 k 1 k · ··· · ··· with an explicit estimator R(a ,...,a ), are established. Here, a ,...,a are assumed to 1 k 1 k belong to the same Cartan subgroupof G. Konstantoulasshowsthat this estimate holds for aL2-densesubspaceoffunctions,butitseemsthatthemethodofproofcannotexplicatethis subspace. Inparticular,theconstantC(φ ,...,φ )isnotexplicit. 1 k We also note that the estimator R(a ,...,a ) in (1.5) is quite different from our estimator 1 k MrestrictedtotheCartansubgroup. Indeed,itmayhappenthattheestimatorR(a ,...,a ) 1 k doesnottendtozeroforsomesequencesuchthata−1a foralli = j. Inparticular,(1.5) i j → 6 doesnotimplymixingoforderkalongtheCartansubgroup. We stress that the validity for general elements g1,...,gk∞ G of our mixing estimate in ∈ Theorem1.1iscrucialforthecombinatorialapplicationthatwediscussinSection1.3below. We note that the strong spectral gap assumption in Theorem 1.1 is known to hold in a number of cases. For instance, if some simple factor G of G has rank at least two, G acts 0 0 ergodicallyonX,itfollowsfromKazhdanproperty(T)thattherepresentationofG onL2(X) 0 0 is isolated from the trivial representation. Another important case is when L is a connected semisimpleLiegroupwithfinitecentreandnocompactfactors,andΓ isanirreduciblelattice in L. The action of L on X = Γ\L has strong spectral gap (see [22, p. 285]), and given any closed connected semisimple subgroup G of L, the action of G on X also has strong spectral gap(see[29]). 4 MICHAELBJÖRKLUND,MANFREDEINSIEDLER,ANDALEXANDERGORODNIK Itisnotdifficulttoseethatthecorrelationsoforderkcanbeinterpretedasintegralsagainst theprobability measure m , supportedon thediagonal ∆ (X)in Xk, which is thepush- ∆k(X) k forwardofmunderthediagonalmapX ∆ (X) Xk. Wenotethatthemeasurem is → k ⊂ ∆k(X) invariant undertheactionofthediagonalsubgroup∆ (G)ofGk,anditsprojectionstoeach k of the factors of Xk equal m. More generally, we say that a probability measure ξ on Xk is a k-coupling of (X,m) if its marginals (the push-forwards of ξ onto the factors of Xk) are all equaltom. WegeneralizeTheorem1.1togeneral∆ (G)-invariantk-couplingsof(X,m)asfollows. k Theorem 1.3 (Uniform exponential mixing of all orders for Lie groups). Let G,X,m be as in Theorem 1.1. Then,for every k > 2andforall sufficientlylarge d,there exists δ = δ(k,d) > 0such thatforevery∆ (G)-invariantcouplingξof(X,m),forallφ ,...,φ C (X),andg ,...,g G, k 1 k ∈ c 1 k ∈ wehave ∞ |ξ((g φ ) (g φ ))−m(φ ) m(φ )| M(g )−δS (φ ) S (φ ), 1 1 k k 1 k d,k [k] d 1 d k · ⊗···⊗ · ··· ≪ ··· whereg = (g ,...,g )andMisasinTheorem1.1. Inparticular,theaboveboundisuniformover [k] 1 k all∆ (G)-invariantk-couplingsξof(X,m). k TheproofsofTheorem1.1andTheorem1.3willbegiveninSection2. 1.3. Anapplication: Approximateconfigurationsinlattices It was realized by Furstenbergin his proofof Szemeredi theorem[11] that the analysis of higherordercorrelationsofdynamicalsystemscanleadtodeepcombinatorialresults. Subse- quentdevelopmentsoftheseideashaveallowedtoproveanumberoffar-reachingtheorems concerningtheexistenceoffiniteconfigurationsin"large"sets. Forinstance,wementionthe works of Furstenberg, Katznelson and Weiss [12] and Ziegler [42] which show that for any given subset Ω Rn of positive upper density, for any given k-tuple (v ,...,v ) (Rn)k, 1 k ⊂ ∈ for all sufficiently large dilations t and for all ε > 0 there exist a k-tuple (ω ,...,ω ) Ωk 1 k ∈ andanisometryIofRn suchthat d(tv ,I(ω )) < ε foralli = 1,...,k. i i In other words, the set Ω must contain an approximate isometric copy of any sufficiently dilatedconfiguration. Inparticular, itfollowsfromthisresultthatgivenanylattice Λin Rn, any k-tuple (v ,...,v ) (Rn)k, for all sufficiently large t and for all ε > 0, there exist a 1 k ∈ k-tuple(z ,...,z ) Λk andanisometryIofRn suchthat 1 k ∈ d(tv ,I(z )) < ε foralli = 1,...,k. i i It would be interesting to investigate whether an analogue of this statement holds for other locally compact groups and whether such a result can be made effective in terms of t. Here weaddressthesequestionsforlatticesinsemisimpleLiegroups. To illustrate our general result, let us consider a Fuchsian group Γ Isom(H2) of finite ⊂ covolume. For fixed v H2, we consider a discrete subset Γv of the hyperbolic plane H2. 0 0 ∈ Howrichisthesetofk-tuple(z ,...,z )withz Γv ? Tomakethisquestionmoreprecise, 1 k i 0 ∈ wedefine,forak-tuple(v ,...,v ) H2,itswidthby 1 k ∈ w(v ,...,v ) = mind(v ,v ). 1 k i j i6=j QUANTITATIVEMULTIPLEMIXING 5 We show in Section 1.3 below that for every k > 2, there are constants c ,ε > 0 with the k k propertythatif(v ,...,v ) (H2)k satisfies 1 k ∈ w(v ,...,v ) >c log(1/ε) (1.6) 1 k k forsome0 < ε< ε ,thenthereisak-tuple(z ,...,z ) (Γv )k andanisometryg PSL (R) k 1 k 0 2 ∈ ∈ suchthat d(v ,gz )< ε foralli = 1,...,k. i i Let us make this result more explicit in the case k = 2. We note that this essentially reduces toanalyzingthedistanceset D:= {d(γv ,v ): γ Γ}. 0 0 ∈ Forexample,whenΓ = PSL (Z)andv = i H2,wehave 2 o ∈ D= cosh−12(a2 +b2+c2+d2)/4 : ad−bc= 1, a,b,c,d Z . ∈ (cid:10) (cid:11) To prove our result in this special case (k = 2), one needs to show that there exists c > 0 2 suchthatforeveryε > 0, theintersectionD [c log(1/ε)), ) isε-densein[c log(1/ε), ). 2 2 ∩ Wenotethatin thiscase,usingthefact thatsetofdistancesiscontainedincosh−1(N)/4,itis nothardtocheckthat(1.6)cannotbereplacedbyacondition∞oftheformw(v1,...,vk) > σ∞(ε) withσ(ε) = o(log(1/ε))asε 0+. → Letusnowformulateourapplicationinitsfullgenerality. LetGbeaconnectedsemisimple Lie group G with finite centre and without compact factors, equipped with a left-invariant Riemannian distance ρ on G which is bi-invariant under a fixed maximal compact sub- G group. ForanyirreduciblelatticeΓ < G,weproveinSection3: Corollary 1.4. For every k > 2, there exist c ,ε > 0 such that given any (g ,...,g ) Gk k k 1 k ∈ satisfying w(g ,...,g ):= minρ (g ,g ) > c log(1/ε) 1 k G i j k i6=j forsome0 < ε< ε ,thereexistak-tuple(γ ,...,γ ) Γk andg Gsuchthat k 1 k ∈ ∈ ρ (g ,gγ )< ε foralli = 1,...,k. G i i 1.4. S-algebraicgroups TheresultsofSection1.2canbeextendedtoactionsofS-algebraicsemisimplegroups. Let G GL be a simply connected absolutely simple algebraic group defined over a number n ⊂ fieldF. WedenotebyV thesetofplacesofF,andforv V wewriteF forthecompletion F F v ∈ ofFwithrespecttothenorm| | . LetG = G(F ). WefixafinitesubsetSofV andconsider v v v F · thegroup G := G . (1.7) v v∈S Y Let S = S S where S and S denote the subsets of the Archimedean and the non- f f ⊔ Archimedeanplacesrespectively. Weset ∞ ∞ G := G and G := G , v f v vY∈S vY∈Sf ∞ sothatG = G G . f ∞ × ∞ 6 MICHAELBJÖRKLUND,MANFREDEINSIEDLER,ANDALEXANDERGORODNIK Let us consider a measure-preserving action of G on a probability space (X,m). We then obtain a unitary representationof Gon thespace L2(X). Given a compact opensubgroupU ofG , wedenotebyC (X)U thesubalgebraofL2(X)consistingofthoseU-invariant vectors f inL2(X,m)whicharesmoothwithrespecttotheactionofG . ∞ We say that the action of G on (X,m) has strong spectral gap if the representation of each noncompactfactor G with v S onL2(X)is isolatedfromt∞hetrivial representation. Inthis v ∈ 0 situation there are quantitative bounds on matrix coefficients of C (X)U analogous to (1.2). Inparticular, we referto theworksofBorel, Wallach [4], Oh [31], Clozel, Oh, Ullmo [5], and ∞ Gorodnik, Maucourant, Oh [13] where such such bounds over non-Archimedean fields are discussed. Foreveryv S,letusfixanormonM (F )anddefinetheheightfunctiononGby n v ∈ H(g):= g forg = (g ) G. v v v v∈S k k ∈ v∈S Y OnecancheckthatHisaproperfunctiononG. Withthisnotation,thereexistsδ > 0suchthat forallsufficientlylarged,foreverycompactopensubgroupUofG ,forallφ ,φ C (X)U, f 1 2 ∈ andg G, ∈ ∞ |m((g φ )φ )−m(φ )m(φ )| H(g)−δ Cd φ Cd φ . (1.8) · 1 2 1 2 ≪d,U k G 1k2k G 2k2 This estimate can be established as in the proof of [13, Theorem 3.27] from the bounds for ∞ ∞ representations of the local factors G . In this paper, we prove an analogous estimate for v correlationsofhigherorder. We consider a jointly continuous of G on a locally compact Hausdorff space X, which we assumeis equippedwith aG-invariant probability Borelmeasure m. Given a compact open subgroup U < G , we denote by C (X)U the sub-algebra of C (X) consisting of those U- f c c invariant functions which are smooth with respect to the action of G . We write C (X) for ∞ c the union of these sub-algebras as U-ranges over all compact open subgroups of G . One ∞f readilychecksthatthisisagainasub-algebraofC (X). ∞ c We assume that thereis a family (S ), with d N, of normson C (X)U which satisfythe d ∈ c followingproperties: ∞ N1. Forallsufficientlylarged,anycompactopensubgroupUofG ,andφ C (X)U, f ∈ c φ d,U Sd(φ). ∞ (1.9) k k ≪ N2. For all sufficiently large d, any compact opensubgroupUofG , for all φ C (X)U, ∞ f ∈ c andg G withv S , ∈ v ∈ ∞ φ−g φ ρ (g,e )S (φ), (1.10) k ∞ · k ≪d,U Gv Gv d whereρ denotesaleft-invariantRiemannianmetriconG . Gv ∞ v N2′. For all sufficiently large d, any compact opensubgroupUofG , for all φ C (X)U, f c ∈ andg G withv S , ∈ v ∈ f ∞ φ−g φ Ad(g)−id S (φ), (1.11) d,U d k · k ≪ k k where denotestheoperatornormonEnd(Lie(G ))forafixedchoiceofanormon v k·k ∞ Lie(G ). v N3. For all sufficiently large d, thereexists σ = σ(d) > 0 such that for any compact open subgroupUofG ,φ C (X)U,andg G, f c ∈ ∈ Sd∞(g φ) d,U Ad(g) σSd(φ). (1.12) · ≪ k k QUANTITATIVEMULTIPLEMIXING 7 N4. Thereexistsr > 0suchthatforallsufficientlylarged,anycompactopensubgroupU ofG ,andφ ,φ C (X)U, f 1 2 ∈ c S∞ (φ φ ) S (φ )S (φ ). (1.13) d 1 2 d,U d+r 1 d+r 2 ≪ Such collections of norms can constructed on finite-volume homogeneous spaces of S- algebraicgroups(see,forinstance,[9]and[10,AppendixA]). WecannowformulatethefollowingS-adicgeneralizationsofTheorems1.1and1.7. They willbeprovedinSection4. Theorem1.5(ExponentialmixingofallordersforS-algebraicgroups). LetGbeanS-algebraic group as in (1.7) which acts continuously and in a measure-preserving fashion on a locally compact HausdorffspaceXequippedwithaBorelprobability measurem. We assume that X is equipped with a family of norms S satisfying Properties N1–N4, and that d thereexistsδ > 0suchthatforallsufficientlylarged,foreverycompactopensubgroupUofG ,for 2 f allφ ,φ C (X)U,andg G,wehave 1 2 ∈ c ∈ ∞ |m((g φ )φ )−m(φ )m(φ )| H(g)−δ2S (φ )S (φ ). 1 2 1 2 d,U d 1 d 2 · ≪ Then, for every k > 2andsufficiently large d, there exists δ = δ(k,d,δ ) > 0such that for every 2 compactopensubgroupUofG ,forallφ ,...,φ C (X)U,andg ,...,g G,wehave f 1 k ∈ c 1 k ∈ |m((g φ ) (g φ ))−m(φ ) m(φ )| ∞ H(g ,...,g )−δS (φ ) S (φ ), 1 1 k k 1 k d,U,k 1 k d 1 d k · ··· · ··· ≪ ··· where H(g ,...,g ):= minH(g−1g ). 1 k i j i6=j Theorem 1.6 (Uniform exponential mixing of all orders for S-algebraic groups). Let G,X,m beasin Theorem 1.5. Then,for every k > 2andsufficiently large d, there exists δ = δ(k,d,δ ) > 0 2 suchthatforeverycompactopensubgroupUofG ,forevery∆ (G)-invariant couplingξof(X,m), f k forallφ ,...,φ C (X)U,andg ,...,g G,wehave 1 k ∈ c 1 k ∈ |ξ((g φ ) (∞g φ ))−m(φ ) m(φ )| H(g ,...,g )−δS (φ ) S (φ ). 1 1 k k 1 k d,U,k 1 k d 1 d k · ⊗···⊗ · ··· ≪ ··· Inparticular, theaboveboundisuniformoverall∆ (G)-invariantk-couplingsξof(X,m). k WestressthattheuniformityinTheorem1.6willbecrucialinouranalysisofhigherorder correlationsforactionsofadelegroupsinthenextsection. 1.5. Adelegroups LetG GL beasimplyconnectedabsolutesimplealgebraicgroupdefinedoveranum- n ⊂ berfieldF. LetG(A )bethecorrespondingadelegroupand F X :=G(F)\G(A ) F equipped with the G(A )-invariant Haar probability measure m. For each v V , we fix a F F ∈ norm on M (F ), which for almost all places v should be the max norm. The height v n v k · k functionH: G(A ) R+ isdefinedby F → H(g):= g , forg = (g ) G(A ). (1.14) k kv v v∈VF ∈ F vY∈VF WenotethatHisaproperfunctiononG(A )(see,forinstance,[13,Lemma2.5]). F 8 MICHAELBJÖRKLUND,MANFREDEINSIEDLER,ANDALEXANDERGORODNIK WedenotebyG theproductofG(F )overtheArchimedeanplacesandbyG thegroup v f of finite adeles. We also denoteby U the product of G(F ) over the Archimedean places v v forwhichG(F )is∞compact. GivenasubgroupUofG(A ),wedenotebyC (X)Uthealgebra v F c of compactly supported functions on∞X which are smooth with respect to the action of G ∞ and are U-invariant. When W is a compact open subgroup of G , we introduce a family of f SobolevnormsS onthealgebrasC (X)W (seeSection5). ∞ d,W c ∞ In Section5weprovethefollowing generalization of[13, Theorem3.27] for U -invariant functions. ∞ Theorem 1.7 (Exponential mixing of all ordersfor adele groups). Let G be a simply connected absolutely simple linear algebraic group defined over anumberfieldF. LetX = G(F)\G(A )andde- F notebymHaarprobabilitymeasureonX. WeassumethatGisisotopicoverF forsomeArchimedean v placev V . F ∈ Then, for every k > 2 and sufficiently large d, there exists δ = δ(k,d) > 0 such that for every compact open subgroup W of G , for all U -invariant functions φ ,...,φ C (X)W, and f 1 k c ∈ s ,...,s G(A ),wehave 1 k ∈ F ∞ ∞ |m((s φ ) (s φ ))−m(φ ) m(φ )| H(s ,...,s )−δS (φ ) S (φ ), 1 1 k k 1 k d,W,k 1 k d,W 1 d,W k · ··· · ··· ≪ ··· where H(s ,...,s ):= minH(s−1s ). 1 k i j i6=j SinceHisaproperfunctiononG(A ),Theorem1.7inparticularimpliesthattheactionof F G(A ) on X = G(F)\G(A ) is mixing of all orders. This was previously established in [14], F F but the method in [14] relies on the theory of unipotent flows and does not provide any explicit estimates. In [1], weapply Theorem1.7 to establish effective countingestimatefor the numberofrationalpointsoncertaincompactificationsofvarietiesoftheformGk/∆ (G). k It is quite likely that the assumption in Theorem 1.7 that G is isotopic over F for some v Archimedeanv V canberemoved. Itisneededbecauseourargumentreliesontheresults F ∈ from [9] which are only proved for real homogeneousspaces. Once a S-algebraic version of [9]has beenestablished,Theorem1.7will follow for generalGusingthemethoddeveloped inthispaper. 1.6. Organisationofthepaper InSection2,wediscusshigherordercorrelationsforsemisimpleLiegroups. Inparticular, wereformulateourmainresultsintermsoftheWassersteindistanceforcouplingsandprove theresultsfrom§1.2. Next, we apply the established correlation estimates in Section 3 to deduce Corollary 1.4 regardingexistenceofapproximateconfigurationsinlatticesubgroups. In Section 4 we analyse higher order correlations for S-algebraic groups, and in Section 5 for adele groups. The proofs in Sections 2 and 4 rely on a general inductive estimate for couplings(Proposition7.2)whichisestablishedinSection7. It will become apparent in Section 7 that our method can be applied more generally to studycouplingswhich are invariant underaflowsatisfyingsuitableregularity,growth,and mixingproperties. InSection6,wediscussbasicpropertiesoftheWassersteindistancewhich areusedinthepaper. QUANTITATIVEMULTIPLEMIXING 9 2. HIGHER-ORDER CORRELATIONS FOR SEMISIMPLE LIE GROUPS 2.1. Preliminaries LetGbeaconnectedsemisimpleLiegroupwithfinitecenter. WefixaCartansubgroupA ofG. WedenotebyΣ Hom(A,R×)therootsystemwithrespecttotheadjointactionofthe + ⊂ CartansubgroupAontheLiealgebrag = Lie(G). Wethenhavetherootspacedecomposition g = g0+ gα, (2.1) α∈Σ M whereg0 isthecentraliseroftheLiealgebraofAing,and gα := Z g : Ad(a)Z = α(a)Z, foralla A . ∈ ∈ Wefixachoiceofthesubset(cid:8)Σ+ Σofpositiverootsanddenoteby (cid:9) ⊂ A+ := a A: α(a) > 1, forallα Σ+ ∈ ∈ thecorrespondingclosedpositiv(cid:8)eWeylchamber in A. Thereexist(cid:9)samaximal compact sub- groupKofGsuchthattheCartandecomposition G = KA+K (2.2) holds. Itisastandardfact(seee.g. [15,Ch.9])thatifg = k a k fork ,k Kanda A+, 1 g 2 1 2 g ∈ ∈ thenthecomponenta isunique. Wecallthecomponenta theCartanprojection ofg. g g Let , be an Ad(K)-invariant inner product on g, and denote by the corresponding h· ·i k·k normong. Letρ denotetheleft-invariant distancefunctiononGinducedbytheRiemann- G ianmetriccorrespondingto , . Wenotethatρ isbi-K-invariant. G h· ·i Defineasub-multiplicativefunction onGby op k·k g := max Ad(g)Z : Z g with Z = 1 . (2.3) op k k k k ∈ k k We note that since G is semisimpl(cid:8)e, every transformation Ad(g) sat(cid:9)isfies det(Ad(g)) = 1, so that it has at least one eigenvalue whose absolute value is greater or equal to one. This impliesthat g > 1 forallg G. op k k ∈ Thefollowing lemmasummarizes thebasic propertiesofthefunctionsρ and that G op k·k willbeusedintheproofsofthemaintheorems. Lemma2.1. (i) Foreveryg G, ∈ g = max α(a ), op g k k α∈Σ+ wherea A+ denotestheCartanprojection oftheelementg. g ∈ (ii) Foreveryg G,thereexistsZ gsuchthatAd(Z)isnilpotent, Z = 1,and ∈ ∈ k k g = Ad(g)Z . op k k k k (iii) Thereexistconstantsc > 1andc > 0suchthat 1 2 c−1log g −c 6 ρ (g,e ) 6 c log g +c 1 k kop 2 G G 1 k kop 2 forallg G. ∈ (iv) Thereisaconstantc > 1suchthatforalleveryX gsuchthatAd(X)isnilpotent, 3 ∈ c−1 max(1, X ) 6 exp(X) 6 c max(1, X )dim(G). 3 k k k kop 3 k k 10 MICHAELBJÖRKLUND,MANFREDEINSIEDLER,ANDALEXANDERGORODNIK In the proof of (iv), we will use the following lemma. Since we shall later need a p-adic versionofthislemma,weformulateitmoregenerally. Lemma 2.2. Fix a norm on M (K), where K is a locally compact normed field. Then there exists a n constantc > 0suchthatforeverynilpotentmatrixX M (K), 0 n ∈ exp(X) >c X . 0 k k k k Proof. We fix a normon Kn. Since all the norms on M (K) are equivalent, we may, without n lossofgenerality,assumethatthenorminthelemmasatisfies Av 6 A v foreveryA M (K)andv Kn. (2.4) n k k k kk k ∈ ∈ ForanilpotentmatrixX,weset c(X) := max{ Xv : v Kn suchthat v = 1andX2v= 0}. k k ∈ k k GivenanynilpotentX M (K)andv Kn suchthat v = 1andX2v = 0,wehave n ∈ ∈ k k exp(X) > exp(X)v = v+Xv > Xv −1. k k k k k k k k Hence, exp(X) > c(X)−1. k k Weclaimthat inf{c(X) : X M (K)isnilpotentwith X = 1}> 0. (2.5) n ∈ k k Suppose that (2.5) fails. Since the function c is continuous, the infimum is achieved, and thereexistsnilpotentX M (K)with X = 1suchthatc(X) = 0. ItfollowsthatifX2v = 0, n ∈ k k for some v Kn, then Xv = 0 as well, i.e., ker(X2) = ker(X). By the same token, we get ∈ ker(Xℓ+1) = ker(Xℓ)foreveryl > 1. SinceXisnilpotent,andthusXn = 0, weconcludethat we must have X = 0, but this contradicts our assumption that X = 1. This contradiction k k showsthat(2.5)holds. Hence,thereexistsc′ > 0suchthatforeverynilpotentX, 0 exp(X) > c′ X −1. k k 0k k Inparticular, exp(X) > c′/2 X whenever X >2/c′. k k 0 k k k k 0 On theotherhand, it is clear that exp(X) −1 X is boundedfromabove forall nilpotent k k k k matricesXsuchthat X 6 2/c′. Combiningthesetwoboundscompletestheproof. (cid:3) k k 0 ProofofLemma2.1. Since the norm on g is Ad(K)-invariant, it is clear that g = a . op g op k k k k Wenotethattherootdecomposition(2.1)isorthogonal. DecomposinganelementY gwith ∈ respectto(2.1),weobtainthatfora A+, ∈ Ad(a)Y 2 = α(a)2 Y 2 6 max α(a)2 Y 2 (2.6) α k k k k α∈Σ∪{0} k k α∈XΣ∪{0} (cid:18) (cid:19) = max α(a)2 Y 2. α∈Σ+ k k (cid:18) (cid:19) Moreover, if we choose α0 Σ+ such that α0(a) = maxα∈Σ+α(a) and Y gα0, then the ∈ ∈ equality in (2.6) holds. Hence, we deduce that for every g G, there exists Y contained in ∈ singlerootspacesuchthat Y = 1and k k g = a = Ad(a )Y = maxα(a ) = max α(a ). (2.7) op g op g g g k k k k k k α∈Σ α∈Σ+ Thisproves(i).

Quantitative multiple mixing

Similar Quantitative multiple mixing