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QUANTITATIVE MULTIPLE RECURRENCE FOR TWO AND THREE TRANSFORMATIONS 7 1 SEBASTIAŃDONOSOANDWENBOSUN 0 2 Abstract. Weprovidevariouscounterexamplesforquantitativemulti- n a plerecurrenceproblemsforsystemswithmorethanonetransformation. J Weshowthat 7 Thereexistsanergodicsystem(X, ,µ,T ,T )withtwocommut- 1 2 • X 2 ing transformations such that for every 0 < ℓ < 4, there exists A suchthat ] ∈X S µ(A T nA T nA)<µ(A)ℓ foreveryn,0; D ∩ 1− ∩ 2− Thereexistsanergodicsystem(X, ,µ,T ,T ,T )withthreecom- . • X 1 2 3 h muting transformations such that for every ℓ > 0, there exists t A suchthat a ∈X m µ(A T nA T nA T nA)<µ(A)ℓ foreveryn,0; ∩ 1− ∩ 2− ∩ 3− [ Thereexistsanergodicsystem(X, ,µ,T1,T2)withtwotransfor- • X 1 mations generating a 2-step nilpotent group such that for every v ℓ>0,thereexistsA suchthat 9 ∈X 3 µ(A∩T1−nA∩T2−nA)<µ(A)ℓ foreveryn,0. 1 8 0 . 1 1. Introduction 0 7 1.1. Quantitativerecurrence. ThePoincare´recurrencetheoremstatesthat 1 : for every measure preserving system (X, ,µ,T) and every A with v X ∈ X i µ(A) > 0, theset X n Z : µ(A T nA) > 0 − r { ∈ ∩ } a isinfinite. AquantitativeversionofitwasprovidedbyKhintchine[8],who showed that for every ǫ > 0, the set n N : µ(A T nA) > µ(A)2 ǫ is − { ∈ ∩ − } syndetic,meaningthatit hasboundedgaps. Multiplerecurrence problems refers to theones concerning the behavior of the set A T nA T nA. In the case T = Ti for an ergodic transfor- ∩ 1− ···∩ d− i mationT, Furstenberg [6]showedthattheset n Z : µ(A T nA T dnA) > 0 − − { ∈ ∩ ···∩ } 2010MathematicsSubjectClassification. Primary:37A30. Key words and phrases. Poincare´ recurrence, multiple recurrence, commuting transformations. ThefirstauthorissupportedbyFondecytIniciaciońenInvestigaciońgrant11160061. 1 2 SEBASTIAŃDONOSOANDWENBOSUN isinfinite. ThisresultisnowknownastheFurstenbergMultipleRecurrence Theorem. The quantitative version of the multiple recurrence problems concerns not only the positivity of the measure of the set A T nA T nA, but ∩ 1− ··· ∩ d− alsohowfarfrom 0 thismeasureis. Moreprecisely,we ask: Question1.1. Let(X, ,µ,T ,...,T )beameasurepreservingsystemand 1 d X F: [0,1] R beanon-negativefunction. Is theset 0 → ≥ n Z : µ(A T nA T nA T nA) F(µ(A)) { ∈ ∩ 1− ∩ 2− ∩···∩ d− ≥ } syndeticforall A ? ∈ X IftheanswertothequestionisaffirmativeforsomeT ,...,T and F,we 1 d say that F is good for (T ,...,T ). Based on the result of Khintchine [8] 1 d statingthat F(x) = x2 ǫ isgoodfor(T)(withasingleterm),anaturalcon- jecturewouldbethatt−hefunction F(x) = xd+1 ǫ isgoodfor(T ,...,T ). 1 d The case T = Ti was solved by Bergelso−n, Host and Kra [3]. They i showed that if (X, ,µ,T) is ergodic, then F(x) = xd+1 ǫ is good for (T,T2,...,Td)foraXllǫ > 0ford = 2or3. Theyalsoshowe−dtwosurprising phenomena. First, the hypothesis of ergodicity cannot be removed: there existsa non-ergodic system(X, ,µ,T)such that F(x) = xℓ is not good for (T,T2) for all ℓ > 0. SecondlyX, if d 4, then F(x) = xℓ is not good for ≥ (T,T2,...,Td)forall ℓ > 0. In[7],FurstenbergandKatznelsonprovedamultiplerecurrencetheorem for commuting transformations. For two commuting transformations, its quantitative study was done by Chu [4] who proved that for every system (X, ,µ,T ,T ) with two commuting transformations T and T (meaning 1 2 1 2 thatXT T = T T ), ergodic for T ,T , and every ǫ > 0, F(x) = x4 ǫ is 1 2 2 1 1 2 h i − good for (T ,T ). Here it is worth to stress that the exponent for (T ,T ) 1 2 1 2 is 4 while the exponent for (T,T2) is 3. Indeed, in the same paper, Chu constructed an exampleshowing that F(x) = x3 is not good for (T ,T ). In 1 2 a later work, Chu and Zorin-Kranich [5] improved this example, showing that F(x) = x3.19 isnotgoodfor(T ,T ). 1 2 Inthispaper,westudythebestcomponentℓ forwhich F(x) = xℓ isgood for (T ,...,T ) in an ergodic system (X, ,µ,T ,...,T ) with commuting 1 d 1 d X transformations. The result of Chu and Zorin-Kranich [5] suggested that the largest ℓ not good for (T ,T ) is between 3.19 and 4. We show that ℓ 1 2 can besufficientlycloseto 4: Theorem 1.2. There exists a measure preserving system (X, ,µ,T ,T ) 1 2 X with commuting transformations T and T , ergodic for T ,T such that 1 2 1 2 forevery0 < ℓ < 4, F(x) = xℓ is notgoodfor(T ,T ), i.e.hthereiexistsa set 1 2 A suchthat ∈ X µ(A T nA T nA) < µ(A)ℓ ∩ 1− ∩ 2− QUANTITATIVEMULTIPLERECURRENCEFORTWOAND THREETRANSFORMATIONS3 foreveryn , 0. Churaisedanotherquestion[4]onwhetherF(x) = xℓisgoodfor(T ,T ,...,T ) 1 2 d inanergodicsystemwithdcommutingtransformationsford 3. Weshow ≥ thatthisisnotthecase: Theorem 1.3. There exists a measure preserving system (X,µ,T ,T ,T ) 1 2 3 withthreecommutingtransformationsT andT andT ,ergodicfor T ,T ,T 1 2 3 1 2 3 such that for every ℓ > 0, F(x) = xℓ is not good for (T ,T ,T ), i.eh. there i 1 2 3 existsa subset A suchthat ∈ X µ(A T nA T nA T nA) < µ(A)ℓ ∩ 1− ∩ 2− ∩ 3− foreveryn , 0. If we relax the condition of commutativity of the transformations, the natural condition to look at is nilpotency. Outside the abelian category, we show that there is no polynomial quantitative recurrence even for two transformationsT andT spanninga2-stepnilpotentgroup. Weshow 1 2 Theorem 1.4. There exists a measure preserving system (X, ,µ,T ,T ) 1 2 X such that T and T generate a 2-step nilpotent group T ,T , whose ac- 1 2 1 2 tions is ergodic and such that for every ℓ > 0, F(x) =hxℓ is niot good for (T ,T ), i.e. thereexistssubset A suchthat 1 2 ∈ X µ(A T nA T nA) < µ(A)ℓ ∩ 1− ∩ 2− foreveryn , 0. 1.2. Notationandconventions. Ameasurepreservingsystem(orasystem for short) is a tuple (X, ,µ,T ,...,T ), where (X, ,µ) is a probability 1 d X X spaceand T ,...,T : X X are actionssuch that forall A ,1 i d, 1 d T 1A andµ(T 1A) =→µ(A). Weuse T ,T ,...,T tode∈noXteth≤egr≤oup i− ∈ X i− h 1 2 di spanned by the transformations T ,...,T . We say that X is ergodic for 1 d T ,...,T ifA ,T 1A = A forall1 i d impliesthatµ(A) = 0or1. h 1 di ∈ X i− ≤ ≤ Forapositiveintegernumber N, thesubset 1,...,N isdenoted by[N]. { } 2. Combinatorial constructions In this section we study subsets of N2 and N3 satisfying special combinatorial conditions that help us construct the counter examples we need. TheconstructionofsuchsetsisinspiredbythemethodsusedbySalemand Spencer [11] and Behrend [2] in building “large” subsets of [N] with no arithmetic progressions of length 3. The ways to make use of special sub- setsinTheorems1.2,1.3and1.4aremotivatedbytheexamplesconstructed inBergelson, Hostand Kra[3]and Chu [4]. Weremarkthatthecombinatorialpropertiesstudiedinthissectionareof independentinterest. 4 SEBASTIAŃDONOSOANDWENBOSUN 2.1. Corner-freesubsetsofN2. Thefirstcombinatorialproblemwestudy ishowlargeasubsetΛ [N]2 can bewithouta“corner”. ⊆ Definition2.1. WesaythatasetΛ [N]2 iscorner-freeif(x,y),(x ,y)and ′ (x,y ) Λ and x y = x y impli⊆esthat x = x and y = y . ′ ′ ′ ′ ′ ∈ − − Wehave Theorem2.2. Letν(N)denotethelargestcardinalityofcorner-freesubsets of[N]2. Then ν(N) > N2−4lolgoglo2g+Nǫ as N forall ǫ > 0. → ∞ It is worth noting that Atjai and Szemere´di [1] had a similar estimate for the largest cardinality of the set Λ [N]2 such that (x,y), (x ,y) and ′ (x,y ) Λ,x > x and x y = x y im⊆plies that x = x and y = y (with ′ ′ ′ ′ ′ ′ ∈ − − an additional but not essential assumption that x > x). Since our proof is ′ different from their method, we write it down for completeness. We thank T.Zieglerforbringingusto thisreference. Proof. Let1 d nbetwoparameterstobechosenlaterandassumethat n isdivisible≪by d2≪. Let Λ betheset ofpoints(x,y) [(2d)n]2 such that the followingconditionholds: expand x = x +x (2d 1∈)+ +x (2d 1)n 1 0 1 n 1 − andy = y +y (2d 1)+ +y (2d 1)n 10 x,y− 2d··2· for0− i −n 1. 0 1 n 1 − i i Consider the n pa−irs o·f·i·nteg−ers (x−,y ),(x≤,y ),..≤.,(x− ,y ).≤De≤fine−Λ 0 0 1 1 n 1 n 1 by(x,y) Λ ifand onlyifamongthepairs(x ,y ),(x ,y−),..−.,(x ,y ), 0 0 1 1 n 1 n 1 there are∈exactly n of them that are equal to (i, j) for all 0 i, j − d −1. d2 ≤ ≤ − RecallforasetS ofcardinalityk andk thatdividesk, k! isthenumber of different patitions of S where each a′tom has exactly(kk/k′)e!kl′ements. Using ′ thisformula, wegetthat n! Λ = . | | ((n )!)d2 d2 We claim that Λ satisfies the properties we are looking for. We first estimate the size of Λ. Set n = d2ω(d), where ω: N N is an increasing → function such that ω(d + 1) ω(d) = O(1), ω(d) and logω(d) 0 as − logd → ∞ logd → d . Forevery N N, pickd N such that → ∞ ∈ ∈ (2.1) (2d 1)d2ω(d) N < (2d +1)(d+1)2ω(d+1). − ≤ By the Stirling formula, if d and n/d2 = ω(d) are large enough, we have that n! nn√2πne n 1 d2n d2d2ω(d) > − = , (( n )!)d2 [(n/d2)n/d2 2π(n/d2)e n/d2]d2 Cd2 ≥ (γn)d2/2 (γω(d))d2/2 d2 − d2 p QUANTITATIVEMULTIPLERECURRENCEFORTWOAND THREETRANSFORMATIONS5 whereγ = 2πC2,andC isaconstantas closeto1 as wewant. So (2.2) N2 (2d+1)2(d+1)2ω(d+1) (2d+1)2(d+1)2ω(d+1)(γω(d))d2/2 log < log < log L !  L   d2d2ω(d)  | |  | |   d2  = 2(d +1)2ω(d+1)log(2d +1) 2d2ω(d)logd+ (logγ+logω(d)) − 2 = d2ω(d)(2log2+o(1)), where in the last step we repeatedly used the properties of ω(d). On the otherhand, by(2.1), wehave logN d2ω(d)log(2d 1) ≥ − and loglogN < 2log(d +1)+logω(d +1)+loglog(2d +1), whichimpliesthat logN (2.3) > d2ω(d)(1/2+o(1)). loglogN Combining(2.2)and(2.3), wehavethat Λ > N2−4lolgoglo2g+Nǫ | | as N forallǫ > 0. → ∞ Nowwe showthat Λ iscorner-free. Supposethat(x,y), (x ,y)and (x,y ) ′ ′ belong to Λ and that x y = x y . Expand w = w + w (2d 1) + ′ ′ 0 1 + w (2d 1)n 1,0 − w ,...,−w d 1 for w = x,y,x ,y .−Since n 1 − 0 n 1 ′ ′ ··· − − ≤ − ≤ − 0 x,y,x ,y d 1 forevery0 i n 1,wehavethatnecessarily ≤ i i ′i ′i ≤ − ≤ ≤ − (2.4) yi − xi = y′i − x′i forall0 i n 1. If y ≤x =≤ (−d 1) for some 0 i n 1, then by the construction of i i Λ,weh−ave x −= d −1andy = 0. B≤y (2≤.4),−y x = y x = (d 1), and so x = d 1iand y−= 0. Thierefore x = x an′id−y′i= y .i− i − − N′iowsu−pposetha′it (d 1) y xi h′i impliiesth′iat(x,y) = (x ,y )for all0 i n 1forso−me−(d ≤1)i −hi ≤d 2. Weshowthiatiy x′i=′ih+1 i i also i≤mpl≤ies−that (x,y) =−(x ,−y ) fo≤r al≤l 0 −i n 1. By the c−onstruction of Λ, the number oif thie pair′is (′ix,y) such≤that≤0 −y x h is the same i i i i ≤ − ≤ as thenumber of the pairs (x,y ) such that 0 y x h, and is the same i ′i ≤ ′i − i ≤ as the number of the pairs (x ,y) such that 0 y x h. Therefore, ′i i ≤ i − ′i ≤ 6 SEBASTIAŃDONOSOANDWENBOSUN by induction hypothesis, if y x = h + 1, we have y x h + 1 and y x h+1. So i − i ′i − i ≥ i − ′i ≥ y x = (y x)+(y x ) (y x) ′i − ′i ′i − i i − ′i − i − i (h+1)+(h+1) (h+1) = h+1 = y x. i i ≥ − − By(2.4), wehavethaty x = y x = h+1 = y x,whichimpliesthat x = x and y = y and w′ie−arie donie−. ′i i − i i ′i i ′i Weconcludethat ν(N) Λ > N2−4lolgoglo2g+Nǫ ≥ | | as N forallǫ > 0. (cid:3) → ∞ 2.2. ThreepointfreesubsetsofN3. Westudyanothercombinatorialprob- leminthissection. Definition 2.3. Let Λ be asubset of[N]3. We say that Λ is three pointfree if(x,y,z ),(x,y ,z),(x ,y,z) Λ impliesthat x = x ,y = y ,z = z . ′ ′ ′ ′ ′ ′ ∈ In particular, (x,y,z ) and (x,y,z) Λ implies that z = z . so Λ contains ′ ′ at most one point on each line parall∈el to Z-axis. Similarly, Λ contains at mostonepointalonganylineparallelto the X orY-axis. Remark 2.4. To a three point free set Λ [N]3 we can associate a N N matrixA(Λ) = (a ) . Todoso,weloo⊆kattheline (i, j,k): k [N]×. If i,j i,j [N] thereisapointofΛin∈ suchalineweseta tobetheu{niqueinteg∈erin}[N] i,j suchthat(i, j,a ) Λ. Ifthereisnopointinsuchalinewejustseta = 0. i,j i,j IfΛ isathreepoin∈tfree set,thematrix A(Λ)has thefollowingproperties: (1) For every k [N], k appears at most once in each row and each columnof A(∈Λ). (2) Foreveryk [N], ifa = k and a = k thena = a = 0. i,j i,j i,j i,j ∈ ′ ′ ′ ′ Conversely, if A is a N N matrix that satisfies conditions (1) and (2), then there is a three point f×ree set Λ [N]3 such that A = A(Λ). The set Λ is just (i, j,a ) : a , 0 and note t⊆hat the cardinality of Λ is the number i,j i,j ofnon-{zero entries of A(Λ}). The combinatorial problem we study is how large such a set three point free set can be. It is clear that if Λ [N]3 is a three point free set, then Λ N2. Weshowthatin fact Λ can⊆besufficiently“close”to N2: | | ≤ | | Theorem 2.5. Let w(N) denote the largest cardinalityof a three point free subsetof[N]3. Then w(N) > N2−4lologglo2g+N2ǫ as N forall ǫ > 0. → ∞ QUANTITATIVEMULTIPLERECURRENCEFORTWOAND THREETRANSFORMATIONS7 Proof. Let n,d, N and Λ begivenin theproofofTheorem 2.2. Define V := (x,y,z): (x,y) Λ [N]3. ∈ ⊆ n o and V := (x,y,z) V: x+y+z = s . s ∈ n o Wehavethat forbigenough N, 3N 3 − Vs = V = N N2−4lolgoglo2g+Nǫ. | | | | · Xs=0 So thereexists0 s 3N 3 such that ≤ ≤ − Vs |V| = N2−4lolgoglo2g+Nǫ/3 > N2−4lologglo2g+N2ǫ | | ≥ 3N provided N is large enough. We verify that V is three point free. Suppose s that(x,y,z ), (x,y ,z)and (x ,y,z)belongto V . Then in particularwe have ′ ′ ′ s that (x,y), (x,y ) and (x ,y) belong to Λ and s x y = z = s x y. So ′ ′ ′ ′ x x = y y. SinceΛ iscorner-free, wehave−tha−t x = x and−y = y−. This ′ ′ ′ ′ im−pliesthat−z = s x y = s x y = zand weconcludethatV isthree ′ ′ s − − − − pointfree. It followsimmediatelythat w(N) Vs N2−4lologglo2g+N2ǫ ≥ | | ≥ providedthat N is largeenough. (cid:3) 3. Nilsystems andaffine nilsystems In all that follows, we make use of the class of nilsystems, specially of affinenilsystemsand webriefly introducethem. 3.1. Affine nilsystems with a single transformation. Let G be a group. For a,b G, [a,b] := aba 1b 1 denotes the commutator of a and b and − − ∈ for A and B subsets ofG, and [A,B]denotes the group generated by all the commutators [a,b] for a A and b B. The commutator subgroups are defined recursively as G ∈= G and G∈ = [G ,G], j 1. We say that G is 1 j+1 j d-stepnilpotent ifG = 1 . ≥ d+1 Let G a d-step nilpotent{L}ie group and Γ be a discrete a cocompact sub- group ofG. Thecompact manifoldG/Γ is a d-stepnilmanifold. The group G acts onG/Γas left translationsandthereis auniqueprobabilitymeasure µ which invariant under such action (called the Haar measure). A dynam- ical system of the form (G/Γ, (G/Γ),µ,T ,...,T ) is called a nilsystem, 1 n where (G/Γ) is the Borel σ-aBlgebra of G/Γ, and each T is given by the i rotationBby afixed τ G, i.e. T : G/Γ G/Γ,x τ x. i i i ∈ → 7→ 8 SEBASTIAŃDONOSOANDWENBOSUN An important class of such systems are the affine nilsystems. The affine nilsytems for a single transformation are defined as follows. Let α Td and let A be a d d unipotent integer matrix (i.e. (A I)d = 0).∈Let T: Td Td be th×e affine transformation x Ax+α. Le−t G be the group → 7→ oftransformationsof Td generated by A and thetranslationsof Td. That is, every element g G is a map x Aix+β for somei Z and β Td. The group G acts tran∈sitively on Td 7→and we may identify t∈his space w∈ith G/Γ, whereΓisthestabilizerof0,whichconsistsofthepowersofA. Thesystem (Td,µ d,T) is called an affine nilsystem (here µ is the Haar measure on T). ⊗ Propertiessuchas transitivity,minimality,ergodicityand uniqueergodicity are equivalentfor asystemin this class and thiscan bechecked by looking attherotationinducedby αontheprojectionTd/Ker(A I) [10]. − 3.2. Affine nilsystems with several transformations. When we consider different affine transformations T : Td Td, x A x + α, where A is i i i i unipotentforeveryi = 1,...,n,wecans→tillregard7→thesystem(Td, (Td),µ d ⊗ B , T ,...,T ) as a nilsystemas long as the matrices commute. Indeed, letG 1 n be the group of transformations of Td generated by the matrices A ,...,A 1 n and the translations of Td. Then every element g G is a map x A(g)x+β(g), where A(g) = Am1 Amn,m ,...,m Z∈and β(g) Td. 7→ 1 ··· n 1 n ∈ ∈ A simple computation shows that if g ,g G, the commutator [g ,g ] 1 2 1 2 is the map x x+(A(g ) I)β(g )+(A(g )∈ I)β(g ) and thus is a trans- 1 2 2 1 7→ − − lation of Td. In the other hand, if g G and β Td, then [g,β] is the translation x x + (A(g) I)β. It ∈follows that∈the iterated commuta- 7→ − tor [ [[g ,g ],g ] g ] belongs to Td and is contained in the image of 1 2 3 k ··· ··· (A(g ) I) (A(g ) I). Ifkislargeenough,thisproductistrivial. SoGis 3 k a nilpo−tent·L··ie group−. The torus Td can be identified withG/Γ, where Γ is thestabilizerof0,which isthegroupgenerated bythematrices A ,...,A . 1 n Wereferto(Td, (Td),µd,T ,...,T )asanaffinenilsystemwithntransfor- 1 n B mations. It is worth noting that the transformations T and T commute if i j (A I)α = (A I)α inTd. i j j i − − By a theoremfrom Leibman[9], weget: Proposition 3.1. Let (Td, (Td),µd,T ,...,T ) be an affine nilsystem with 1 n B ntransformations. Thenthepropertiesoftransitivity,minimality,ergodicity anduniqueergodicityunder theactionof T ...,T areequivalent. 1 n h i Recall that (Td, (Td),µd,T ,...,T ) is minimal under T ...,T if the 1 n 1 n B h i orbit closure of every x Td under T ...,T is Td (in this paper we do 1 n ∈ h i notneed theconceptsoftransitivityand uniqueergodicity). Werefer to[9] foramodernreference on nilmanifoldsandnilsystems. QUANTITATIVEMULTIPLERECURRENCEFORTWOAND THREETRANSFORMATIONS9 4. Proofs ofthemain theorems Weare nowready to provethemaintheorems. ProofofTheorem 1.2. Let α,β R Q be rationally independent numbers. Let X = T6 withtransformations∈ \ T (x ,x ,x ;y ,y ,y ) = (x +α,x +β,x ;y + x ,y ,y + x + x + x ) 1 1 2 3 1 2 3 1 2 3 1 1 2 3 1 2 3 and T (x ,x ,x ;y ,y ,y ) = (x ,x +β,x +α;y ,y + x ,y + x + x + x ). 2 1 2 3 1 2 3 1 2 3 1 2 3 3 1 2 3 Wehavethat(X, (T6),µ µ µ µ µ µ,T ,T )isanaffinenilsystem 1 2 B ⊗ ⊗ ⊗ ⊗ ⊗ withtwotransformations,whereµ istheHaarmeasureon T. Noticethat T T (x ,x ,x ;y ,y ,y ) = T T (x ,x ,x ;y ,y ,y ) 1 2 1 2 3 1 2 3 2 1 1 2 3 1 2 3 = (x +α,x +2β,x +α;y + x ,y + x ,y + x + x + x +α+β). 1 2 3 1 1 2 3 3 1 2 3 We first claim that the system is minimal and ergodic. To see this, let (x ,x ,x ;y ,y ,y ) and (x ,x ,x ;y ,y ,y ) T6. It is not hard to see that 1 2 3 1 2 3 ′1 ′2 ′3 ′1 ′2 ′3 ∈ (x ,x ;x ,y ,y ,y ) belongs to the orbit closure of (x ,x ,x ;y ,y ,y ) un- ′1 ′2 3 ′1 2 ′3 1 2 3 1 2 3 der the transformation T . We also have that (x ,x ,x ;y ,y ,y ) is in the 1 ′1 ′2 ′3 ′1 ′2 ′3 orbit closure of (x ,x ,x ,y ,y ,y ) under the transformation T (here we ′1 ′2 3 ′1 2 ′3 2 use the fact that for fixed x , the transformation (x ,x ,y ,y ) (x + 1 2 3 2 3 2 β,x + α,y + x ,y + x + x + x ) is minimal in T4). We conc7→lude that 3 2 3 3 1 2 3 (x ,x ,x ,y ,y ,y )isintheorbitclosureof(x ,x ,x ;y ,y ,y )under T ,T . ′1 ′2 ′3 ′1 ′2 ′3 1 2 3 1 2 3 h 1 2i Sincesincethepointsarearbitrary,thesystemisminimalandhenceergodic byProposition3.1. Let N N to be chosen later and Λ [N]3 be a three point free set. For ∈ ⊆ (a,b,c) [N]3, denote ∈ a a 1 b b 1 c c 1 Q = , + , + , + T3 a,b,c N N 2N × N N 2N × N N 2N ⊆ (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) andlet B = Q , A = T3 B. Then (a,b,c) Λ a,b,c ∪ ∈ × Λ µ µ µ µ µ µ(A) = | | . ⊗ ⊗ ⊗ ⊗ ⊗ 8N3 Ontheotherhand, µ µ µ µ µ µ(A T nA T nA) ⊗ ⊗ ⊗ ⊗ ⊗ ∩ 1− ∩ 2− = 1 (y ,y ,y )1 (y ,y ,y )1 (y ,y ,y )dµ µ(x ,x ,x ;y ,y ,y ), Z B 1 2 3 B ′1 2 ′3 B 1 ′2 ′3 ⊗···⊗ 1 2 3 1 2 3 T6 where n n n y = y +nx + α, y = y +nx + α, y = y +n(x +x +x )+ (α+β). ′1 1 1 2! ′2 2 2 2! ′3 3 1 2 3 2! 10 SEBASTIAŃDONOSOANDWENBOSUN Suppose that the product of the functions inside the integral is nonzero. Then we may assume that (y ,y ,y ) Q , (y ,y ,y ) Q and 1 2 3 ∈ a,b,c ′1 2 ′3 ∈ a′,b,c′ (y ,y ,y ) Q for some (a,b,c), (a ,b,c ) and (a,b ,c ) that belong to1Λ.′2Si′3nce∈Λ ias,b′t,hc′ree point free, we have′ that′a = a , b ′= b′ and c = c . ′ ′ ′ Thisimpliesthat n n n 1 1 nx + α, nx + α, y +n(x + x + x )+ (α+β) ( , ). 1 2 3 1 2 3 2! 2! 2! ∈ −2N 2N Therefore, µ µ µ µ µ µ(A T nA T nA) ⊗ ⊗ ⊗ ⊗ ⊗ ∩ 1− ∩ 2− = 1 (y ,y ,y )1 (y ,y ,y )1 (y ,y ,y )dµ µ(x ,x ,x ;y ,y ,y ) Z B 1 2 3 B ′1 2 ′3 B 1 ′2 ′3 ⊗···⊗ 1 2 3 1 2 3 T6 1 1 Λ = 1 (y ,y ,y )dµ(y )dµ(y )dµ(y ) = | | . N3 Z B 1 2 3 1 2 3 N3 · 8N3 T3 Wehavethatµ µ µ µ µ µ(A T nA T nA) = Λ Λℓ = µ(A)ℓ ⊗ ⊗ ⊗ ⊗ ⊗ ∩ 1− ∩ 2− 8|N|6 ≤ (8|N|3)ℓ ifand onlyif 3 (4.1) ℓ 1+ . ≤ 3+ log(8) log(Λ) | | log(N) − log(N) ByTheorem 2.5, wecan takeΛ ofcardinalitylargerthan N2 ǫ and thusthe − right hand side in (4.1) can be as close to 4 as we want. This finishes the proof. (cid:3) ProofofTheorem 1.4. Letα R Q. Let X = T3 withtransformations ∈ \ T (x,y,z) = (x+α,y+ x,z) 1 and T (x,y,z) = (x+α,y,z+ x). 2 It is an affine nilsystem with two transformations. We first claim that the system (X, (T3),µ µ µ,T ,T ) is minimal and ergodic, where µ is 1 2 B ⊗ ⊗ the Haar measure on T. To see this, take (x ,y ,z ) T3 and note that the 0 0 0 ∈ closureoftheorbitofthispointunderthetransformationT isT T z . 1 0 × ×{ } Let(x,y,z) T3 bean arbitrarypointandnoticethatthispointiscontained ∈ intheorbitclosureof(x ,y,z )underthetransformationT . So theclosure 0 0 2 of the orbit of (x ,y ,z ) under T ,T is T3. We conclude that the system 0 0 0 1 2 h i isminimaland thusergodicbyProposition3.1. It is easy to verify that [T ,T ](x,y,z) = (x,y+α,z α) commuteswith 1 2 − T and T . So T andT generatea 2-stepnilpotentgroup. 1 2 1 2