Multiple ergodic averages in abelian groups and Khintchine type recurrence
MMULTIPLE ERGODIC AVERAGES IN ABELIAN GROUPS ANDKHINTCHINE TYPE RECURRENCE
OR SHALOM
Abstract.
Let G be a countable abelian group. We study ergodic averages associ-ated with arithmetic progressions { ag, bg, ( a + b ) g } for some a, b ∈ Z . Under someassumptions on G , we prove that the universal characteristic factor for these averagesis a factor (Definition 1.15) of a 2-step nilpotent homogeneous space (Theorem 1.18).As an application we derive a Khintchine type recurrence result (Theorem 1.3). Inparticular, we prove that for every countable abelian group G , if a, b ∈ Z are such that aG, bG, ( b − a ) G and ( a + b ) G are of finite index in G , then for every set E ⊂ G andevery ε > { g ∈ G : d ( E ∩ E − ag ∩ E − bg ∩ E − ( a + b ) g ) ≥ d ( E ) − ε } is syndetic. This generalizes previous results for G = Z , G = F ωp and G = (cid:76) p ∈ P F p by Bergelson Host and Kra [4], Bergelson Tao and Ziegler [7] and the author [22],respectively. Introduction
Multiple ergodic averages play an important role in ergodic Ramsey theory. In thecase of Z -actions they were used by Furstenberg [13] to prove Szemer´edi’s theorem [25]about the existence of arbitrary large arithmetic progressions in sets of positive upperBanach density. The goal of this paper is to study the convergence and limit of somemultiple ergodic averages associated with 4-term arithmetic progressions in countableabelian groups. As usual, a G -system X = ( X, B , µ, T g ) is a probability space ( X, B , µ )which is regular and separable modulo null sets, together with an action of a countableabelian group G on X by measure preserving transformations T g : X → X . For a, b ∈ Z ,a Følner sequence Φ N of G and bounded functions f , f , f ∈ L ∞ ( X ) we study themultiple ergodic averages(1.1) E g ∈ Φ N f ( T ag x ) f ( T bg x ) f ( T ( a + b ) g x )where E g ∈ Φ N = | Φ N | (cid:80) g ∈ Φ N . The L -convergence of these averages as N goes to infinity isalready known for all countable nilpotent groups (see Austin [2]). In the case of Z -actions,these averages were studied by Conze and Lesigne [8], [9], [10] and by Furstenberg andWeiss [15] using the theory of characteristic factors (see Definition 1.6). This theory were Date : February 16, 2021. meaning that X is a compact metric space, B is the completion of the σ -algebra of Borel sets, and µ is a Borel measure. a r X i v : . [ m a t h . D S ] F e b OR SHALOM developed further by Host and Kra [18] and Ziegler [27] in order to deduce the convergenceof some multiple ergodic averages associated with Z -actions and by Bergelson Tao andZiegler with F ωp actions [6].This paper is focused on one of the many applications for these structure theoremsassociated with the Khintchine type recurrence. For example, we begin with the followingresult by Bergelson Host and Kra [4]. Theorem 1.1.
Let ( X, B , µ, T ) be an invertible ergodic system. Then, for any measurableset A ∈ B and ε > the set { n ∈ Z : µ ( A ∩ T − n A ∩ T − n A ∩ T − n A ) > µ ( A ) − ε } is syndetic . In [7] Bergelson Tao and Ziegler proved a counter-part to this result associated withthe group F ωp . This was generalized by the author in [22]. We have, Theorem 1.2.
Let P be a countable multiset of primes with < min p ∈ P p and let G = (cid:76) p ∈ P F p . Then for any ergodic G -system X , a measurable set A and ε > the set { g ∈ G : µ ( A ∩ T g A ∩ T g A ∩ T g A ) > µ ( A ) − ε } is syndetic. In this paper we generalize the above for all countable abelian groups, under thefollowing conditions.
Theorem 1.3 (Khintchine type recurrence result for countable abelian groups) . Let G be a countable abelian group and fix a, b ∈ Z . If aG , bG , ( a − b ) G and ( a + b ) G are offinite index in G then for every ergodic G -system ( X, B , µ, T g ) , a measurable set A and ε > the set { g ∈ G : µ ( A ∩ T ag A ∩ T bg A ∩ T ( a + b ) g A ) ≥ µ ( A ) − ε } is syndetic. A direct application of the Furstenberg correspondence principle gives,
Theorem 1.4 (Density result) . Let G be a countable abelian group and a, b ∈ Z suchthat aG , bG , ( b − a ) G and ( a + b ) G are of finite index. Let Φ N be any Følner sequence for G and d Φ be the corresponding upper density. i.e. d Φ ( E ) = lim sup N →∞ | E ∩ Φ N || Φ N | . Thenfor any set E ⊆ G and ε > , the set { g ∈ G : d Φ ( E ∩ E − ag ∩ E − bg ∩ E − ( a + b ) g ) ≥ d Φ ( E ) − ε } is syndetic. Recall that a set A in a group G is syndetic if there exists a finite set C ⊆ G such that A + C = G . RGODIC AVERAGES AND KHINTCHINE RECURRENCE 3
Remark 1.5.
The case of double recurrence. Namely that, { g ∈ G : µ ( A ∩ T ag A ∩ T bg A ) ≥ µ ( A ) − ε } is syndetic, is not covered in this paper. This and a more general version of doublerecurrence can be found in a recent paper by Ackelsberg Bergelson and Best [1].Roughly speaking, we say that a factor of an ergodic system X is characteristic for anergodic average if the limit behavior of the average can be reduced to this factor. Theassumption on the indices of aG , bG , ( b − a ) G and ( a + b ) G in Theorem 1.3 is necessaryto ensure that the systems we study in this paper are characteristic for average (1.1). Definition 1.6 (Characteristic factors) . Let G be a countable abelian group and let X be an ergodic G -system. For k ∈ N and 0 (cid:54) = a , ..., a k ∈ Z , we say that a factor Y ischaracteristic for the tuple ( a g, a g, ..., a k g ) if for every bounded functions f , ..., f k ∈ L ∞ ( X ) and every Følner sequence Φ N of G we have,lim N →∞ (cid:32) E g ∈ Φ N k (cid:89) i =1 T a i g f i − E g ∈ Φ N k (cid:89) i =1 T a i g E ( f i | Y ) (cid:33) = 0in L where E ( f i | Y ) denotes the conditional expectation with respect to the factor Y . Example 1.7. • X is a characteristic factor for any tuple. • The mean ergodic theorem states that the trivial factor is characteristic for ( g ). • It is well known that for any countable abelian group G , the Kronecker factor(the maximal group rotation factor) is a characteristic factor for ( g, g ). • If G = Z then the Kronecker factor is also characteristic for ( ag, bg ) for any0 (cid:54) = a, b ∈ Z , and in [15] Furstenberg and Weiss proved that the Conze-Lesignefactor is characteristic for ( ag, bg, ( a + b ) g ). We will discuss this below.In the case of Z -actions, Host and Kra [18] proved that characteristic factors for thetuple ( g, g, g, ..., kg ) are closely related to an infinite version of the Gowers norms. Definition 1.8 (Gowers Host Kra seminorms) . Let G be a countable abelian group,let X = ( X, B , µ ) be a G -system, let φ ∈ L ∞ ( X ), and let k ≥ (cid:107) φ (cid:107) U k of order k of φ is defined recursively by the formula (cid:107) φ (cid:107) U := lim N →∞ | Φ N | (cid:107) (cid:88) g ∈ Φ N φ ◦ T g (cid:107) L for k = 1, and (cid:107) φ (cid:107) U k := lim N →∞ | Φ kN | (cid:88) g ∈ Φ kN (cid:107) ∆ g φ (cid:107) k − U k − / k for k ≥
1, where φ N , ..., φ kN are arbitrary Følner sequences. OR SHALOM
In the case where G = Z /N Z these seminorms where introduced by Gowers in [16]where he derived quantitative bounds for Szemer´edi’s theorem [25].The Host-Kra factors are defined by the following proposition, Proposition 1.9.
Let G be a countable abelian group, let X be an ergodic G -system,and let k ≥ . Then, there exists a factor Z Let G be a countable abelian group and X be an ergodic G -system.For k ≥ X is a system of order < k if it is isomorphic as a G -system tothe factor Z Convention. For an ergodic G -system X , we call Z < ( X ) the Kronecker factor and Z < ( X ) the C.L factor (named after Conze and Lesigne [8], [9], [10]) and we iden-tify Z < ( X ) with a group rotation (see Definition 5.1). Similarly, if X = Z < ( X ) or X = Z < ( X ) we say that X is a Kronecker system or a C.L system, respectively.It is well known that the Conze-Lesigne factor is an abelian extension of the Kroneckerfactor by an abelian group and a C.L cocycle. We define these notions below, Definition 1.12 (Abelian cohomology) . Let G be a countable abelian group and let X be a G -system and ( U, · ) a compact abelian group. A measurable function ρ : G × X → U is called a cocycle if ρ ( g + g (cid:48) , x ) = ρ ( g, x ) · ρ ( g (cid:48) , T g x ) for every g, g (cid:48) ∈ G and µ -almostevery x ∈ X . The abelian extension of X by the cocycle ρ is defined to be the productspace X × ρ U = ( X × U, B X ⊗ B U , µ X ⊗ µ U , S g )together with the action S g ( x, u ) = ( T g x, ρ ( g, x ) u ). We denote this system by X × ρ U .We note that if two cocycles ρ, ρ (cid:48) : G × X → U are cohomologous, namely if thereexists a measurable map F : X → U such that ρ ( g, x ) /ρ (cid:48) ( g, x ) = ∆ g F ( x ) for all g ∈ G and µ -almost every x ∈ X . Then they define isomorphic group extensions. RGODIC AVERAGES AND KHINTCHINE RECURRENCE 5 Observe that the group U acts on the extension X × U by measure preserving transfor-mations V u ( x, v ) = ( x, uv ). More generally, given an action of a compact abelian group A on a system X , we define V a f ( x ) = f ( ax ) and ∆ a f ( x ) = V a f ( x ) · f ( x ).Below we define the notion of a C.L cocycle with respect to the group A . Definition 1.13 (Conze-Lesigne cocycles) . Let G be a countable abelian group andlet X be an ergodic G -system. Let U and A be compact abelian groups and supposethat A acts on X by measure preserving transformations. We say that the cocycle ρ : G × X → U is a C.L cocycle with respect to A if for every a ∈ A there exist ahomomorphism c a : G → U and a measurable map F a : X → U such that∆ a ρ ( g, x ) = c ( g ) · ∆ g F ( x )for µ -almost every x ∈ X and all g ∈ G .In [15] Furstenberg and Weiss proved, Theorem 1.14 ( Z < ( X ) is an extension of the Kronecker by a C.L cocycle) . Let ( X, B , µ, T ) be an invertible measure preserving system. Then there exists a compact abelian group U and a cocycle ρ : G × Z < ( X ) → U such that Z < ( X ) = Z < ( X ) × ρ U and for every χ ∈ ˆ U , χ ◦ ρ is a C.L cocycle with respect to Z < ( X ) . The Conze-Lesigne factor as a factor of a nilpotent system. We briefly andinformally explain how the methods we use in the proof of Theorem 1.3 differ from theprevious cases for Z and (cid:76) p ∈ P F p (Theorem 1.1 and Theorem 1.2).The main difficulty in the proof of these theorems is to show that the Conze-Lesginefactor admits some nilpotent structure. This nilpotent structure leads to a convenientformula for the limit of average (1.1), which can be used to derive the recurrence result.In the generality of countable abelian groups we only managed to give partial resultsin this direction. More specifically, we show that for any ergodic system ( X, G ) thereexists an extension ( Y, H ) (Definition 1.15) such that the C.L factor of Y has the struc-ture of a 2-step nilpotent homogeneous space. As usual, we reduce the study of thelimit of average (1.1) to the case where the functions are measurable with respect to theConze-Lesigne factor Z < ( X ) (Theorem 2.3). The main difference is that now we haveto pull everything up to the extension Z < ( Y ). Using the nilpotent structure of Z < ( Y )we derive a formula for the limit of some multiple ergodic averages (Theorem 6.1). Thisformula is used to deduce the Khintchine type recurrence result in Theorem 1.3.We begin by introducing a notion of an extension outside of the category of G -systems.Observe, that for a G -system X = ( X, B , µ, T g ), and a countable abelian group H with asurjective homomorphism ϕ : H → G there exists a natural H -action on X by S h = T ϕ ( h ) .This leads to the following definition, Definition 1.15 (Extensions) . Let G and H be countable abelian groups. We say thatthe system Y = ( Y, ( S h ) h ∈ H ) is an extension of ( X, ( T g ) g ∈ G ) if there exists a surjective OR SHALOM homomorphism ϕ : H → G and a factor map π : Y → X such that π ◦ S h = T ϕ ( h ) ◦ π forall h ∈ H . Example 1.16. Let G = Z / Z and let X = {− , } , then G acts on X by T g x = x g .Similarly, let H = Z / Z and Y = {− , − i, i, } then H acts on Y by S h y = y h . Thesystem ( Y, H ) defines an extension of ( X, G ) with respect to the the homomorphism ϕ : H → Gϕ ( h ) = h mod 2and the factor map π : Y → X with π ( y ) = y .In particular, we see from this example that the family of ergodic H -extensions canbe larger than the family of ergodic G -extensions (there is no ergodic G -action on Y ).In example 1.21 below we see another advantage of these extensions.The following group were studied by Conze and Lesigne [8], [9], [10] and generalizedby Host and Kra [18] for systems of order < k , for any k ∈ N (see Definition A.3). Definition 1.17 (C.L group) . Let G be a countable abelian group, let X be a C.L G -system and write X = Z < ( X ) × ρ U for some compact abelian group U . For every s ∈ Z < ( X ) and F : Z < ( X ) → U , let S s,F ∈ Z < ( X ) (cid:110) M ( Z < ( X ) , U ) be the measurepreserving transformation S s,F ( z, u ) = ( sz, F ( z ) u ). The C.L group is given by G ( X ) = { S s,F ∈ Z < ( X ) (cid:110) M ( Z < ( X ) , U ) : ∃ c : G → U such that ∆ s ρ = c · ∆ F } with the natural multiplication S s,f ◦ S t,h = S st,hV t f .Equipped with the topology of convergence in measure G ( X ) is a 2-step nilpotent lo-cally compact polish group.Our main result is the following structure theorem. Theorem 1.18 (Structure Theorem) . Let G be a countable abelian group and let X be an ergodic G -system. Then, there exist an extension ( Y, H ) and a -step nilpotentlocally compact polish group G which acts transitively on Z < ( Y ) by measure preservingtransformations. Moreover, we can take G = G ( Z < ( Y )) . The moreover part in Theorem 1.18 plays an important role in the proof of the Khint-chine type recurrence (Theorem 1.3). More specifically, it is used in the proof of thelimit formula for some multiple ergodic averages (Theorem 6.1).Below is an important remark about the structure of Z < ( Y ) as a homogeneous space. Remark 1.19. In the settings of Theorem 1.18, the system Z < ( Y ) is isomorphic tothe G -system ( G ( Z < ( Y )) / Γ , B , µ, R g ) where Γ is the stabilizer of some x ∈ X , B isthe Borel σ -algebra and µ the Haar measure . In the proof of Theorem 4.1 we show This measure exists because locally compact nilpotent groups are uni-modular. RGODIC AVERAGES AND KHINTCHINE RECURRENCE 7 that Γ is a totally disconnected closed co-compact subgroup of X and there exists ahomomorphism ϕ : G → G ( X ) such that the action R g is given by left multiplication by ϕ ( g ).The factor map π : Y → X , induces a factor ˜ π : Z < ( Y ) → Z < ( X ) and the followingdiagram commutes. ( X , G ) ( Y , H )( Z < ( X ) , G ) ( Z < ( Y ) , H ) ∼ = ( G / Γ , H ) ππ X π Y ˜ π A system ( X, G ) is called a k -step nilsystem if it is isomorphic to a homogeneousspace G / Γ where the homogeneous group is a k -step nilpotent Lie group, Γ is a discreteco-compact subgroup and the action of G is as in the remark above. Conze and Lesigne[8] [9], [10] proved that the C.L factor of an ergodic Z -system is an inverse limit of 2-stepnilsystems. Host and Kra [18] and Ziegler [27] generalized this result, showing that forall k ∈ N , any Z -system of order < k + 1 is an inverse limit of k -step nilsystems.In [26] Ziegler proved a point-wise convergence and limit formula for some multipleergodic averages on a nilsystem (see also [4]). Formally, let X = G / Γ be a k -stepnilsystem and denote by µ G the Haar measure on G . For 1 ≤ r ≤ k + 1, we write G r forthe closed subgroup generated by the commutators of length r in G . Let Γ r = Γ ∩ G r ,and m r be the Haar-measure on the quotient space G r / Γ r . Then, Theorem 1.20. If f , ..., f k +1 ∈ L ∞ ( X ) , and a , ..., a k +1 ∈ N then for µ G -almost every x ∈ G we have, lim N − M →∞ N (cid:88) n = M k +1 (cid:89) i =1 T a i n f i ( x Γ) = (cid:90) G / Γ (cid:90) G / Γ ... (cid:90) G k / Γ k k +1 (cid:89) i =1 f i ( x · k (cid:89) j =1 y ( aij ) j Γ) k +1 (cid:89) i =1 dm i ( y i ) where (cid:0) ba (cid:1) = 0 if a > b . Bergelson Tao and Ziegler [6] proved a counter-part for F ωp -systems; They showed thatany ergodic F ωp -system has the structure of a Weyl system , and proved a similar limitformula for multiple ergodic averages associated with this group [7]. Any Weyl systemof order < k + 1 has the structure of a k -step nilpotent homogeneous space. In the case A Weyl system is a tower of abelian extensions where the cocycles are phase polynomials (seeDefinition 3.1). OR SHALOM of F ωp -actions the homogeneous group and the stabilizer are totally disconnected. Thisstructure theorem was generalized by the author [22] for (cid:76) p ∈ P F p - actions in the specialcase k = 2. More specifically, the Conze-Lesigne factor of an ergodic (cid:76) p ∈ P F p -system isan inverse limit of 2-step nilpotent homogeneous spaces where the homogeneous groupsare finite dimensional.It is natural to ask whether Theorem 1.18 holds without the use of extensions. Thefollowing question is still open. Question. Let G be a countable abelian group. Is it true that every ergodic C.L G -system X is isomorphic to a -step nilpotent homogeneous space? A key component in the proof of Theorem 1.18 is a result about the (point) spectrumof the G action as a unitary operator on X . Let k ≥ 1, we say that a function P : X → S is a phase polynomial of degree < k if for any g , ..., g k ∈ G we have ∆ g ... ∆ g k P = 1and write P X, G )-system there existsan extension ( Y, H ) such that for every element in λ ∈ Spec k ( X ) and n ∈ N there is an n ’th root for the corresponding element in Spec k ( Y ). In the case where k = 1, we show(Proposition 3.14) that Y has a divisible group of eigenfunctions. The following exampleillustrates this phenomena in a simple case. Example 1.21. Let ( X, G ) and ( Y, H ) be as in Example 1.16.The group of eigenfunctions of X consists of the constant functions and constant multi-plications of the embedding χ : X → S , χ ( x ) = x . This group is clearly not divisible,for instance there is no square root for χ . On the other hand, let χ ◦ π be the lift of χ to Y . We see that the eigenfunction τ : Y → S , τ ( x ) = x is a square root of χ ◦ π .This process can be iterated infinitely many times using inverse limits. The result is anextension of X with a divisible group of eigenfunctions. We do this in detail in section3. If ( Y, H ) is a Conze-Lesigne system with a divisible 1-spectrum then G ( Y ) acts tran-sitively on Y (Theorem 4.1). It is natural to ask whether this holds in higher order.Namely, Question. Let k ≥ , let G be a countable abelian group, and let ( X, G ) be an ergodicsystem such that Spec ( X ) , ..., Spec k − ( X ) are divisible. Is it true that Z I would like to thank my adviser Prof. Tamar Ziegler for manyvaluable discussions and suggestions. RGODIC AVERAGES AND KHINTCHINE RECURRENCE 9 The Conze-Lesigne factor is characteristic In this section we prove that under the assumptions in Theorem 1.3, the C.L factoris a characteristic factor for the tuple ( ag, bg, ( a + b ) g ). Our main tool is the van derCorput lemma, (see e.g. [3]). Lemma 2.1 (van der Corput lemma) . Let H be a Hilbert space with inner product (cid:104)· , ·(cid:105) and norm (cid:107) · (cid:107) . Let G be an amenable group, then for any Følner sequence Φ N of G andany bounded sequence { x g } g ∈ G ⊆ H . If lim N →∞ E g ∈ Φ N (cid:104) x g + h , x g (cid:105) exists for every h ∈ G and M ∈ R is such that for any Følner sequence Ψ H , lim sup H →∞ | E g ∈ Ψ H lim N →∞ E g ∈ Φ N (cid:104) x g + h , x g (cid:105) | ≤ M Then, lim sup N →∞ (cid:107) E g ∈ Φ N x g (cid:107) ≤ M In particular, if lim H →∞ E g ∈ Ψ H lim N →∞ E g ∈ Φ N (cid:104) x g + h , x g (cid:105) = 0 then, lim N →∞ E g ∈ Φ N x g = 0 .Proof. Let ε > N and H (cid:107) E g ∈ Φ N x g − E h ∈ Φ H E g ∈ Φ N x g + h (cid:107) < ε We use o ( ε ) to denote a positive quantity that goes to 0 as ε → 0. Since x g is boundedwe have, (cid:107) E g ∈ Φ N x g (cid:107) ≤ E g ∈ Φ N (cid:107) E h ∈ Φ H x g + h (cid:107) + o ( ε )Then, the right hand side becomes E g ∈ Φ N E h ∈ Φ H E h (cid:48) ∈ Φ H (cid:104) x g + h , x g + h (cid:48) (cid:105) + o ( ε )We make a change of variables and change the order of summation. E h (cid:48) ∈ Φ H E h ∈ Φ H E g ∈ Φ N + h (cid:48) (cid:104) x g + h − h (cid:48) , x g (cid:105) + o ( ε )As Φ N is a Følner sequence, taking a limit as N → ∞ we get that for sufficiently large H , the above equals to E h (cid:48) ∈ Φ H E h ∈ Φ H γ h − h (cid:48) + o ( ε )Making a change of variables again this becomes(2.1) E h (cid:48) ∈ Φ H E h ∈ Φ H + h (cid:48) γ h + o ( ε )Let ε > 0, and suppose by contradiction that there exists a sub-sequence H k −→ k →∞ ∞ such that for every k , (cid:12)(cid:12)(cid:12) E h (cid:48) ∈ Φ Hk E h ∈ Φ Hk + h (cid:48) γ h (cid:12)(cid:12)(cid:12) > M + ε Then we can find h (cid:48) k ∈ Φ H K such that (cid:12)(cid:12)(cid:12) E h ∈ Φ Hk + h (cid:48) k γ h (cid:12)(cid:12)(cid:12) > M + ε However, Ψ k = Φ H k + h (cid:48) k is a Følner sequence and we have a contradiction. Therefore thelim sup H →∞ of (2.1) is bounded by M + o ( ε ). As ε > (cid:3) The first application of this lemma is the following result of Furstenberg and Weiss[15]. Lemma 2.2 (The Kronecker factor is characteristic for double averages) . Let G be acountable abelian group and let X be an ergodic G -system. Suppose that a, b ∈ Z aresuch that aG, bG and ( b − a ) G are of index d a , d b and d b − a in G , respectively. Fix f , f ∈ L ∞ ( X ) with (cid:107) f (cid:107) ∞ , (cid:107) f (cid:107) ∞ ≤ and let x g = T ag f · T bg f then lim N →∞ E g ∈ Φ N x g exists and (cid:13)(cid:13)(cid:13) lim N →∞ E g ∈ Φ N x g (cid:13)(cid:13)(cid:13) L ( X ) ≤ d b − a · min { d a · (cid:107) f (cid:107) U ( X ) , d b · (cid:107) f (cid:107) U ( X ) } in L for every Følner sequence Φ N of G .Proof. We follow the argument in [15]. Set x g = T ag f · T bg f then, (cid:104) x g + h , x g (cid:105) = (cid:90) X T ag + ah f · T bg + bh f · T ag f · T bg f dµ Since T ag is measure preserving we have,lim N →∞ E g ∈ Φ N (cid:104) x g + h , x g (cid:105) = lim N →∞ E g ∈ Φ N (cid:90) X ∆ ah f · T ( b − a ) g ∆ bh f dµ By the mean ergodic theorem the limit exists and equals to(2.2) (cid:90) X ∆ ah f P b − a (∆ bh f ) dµ where P b − a is the projection to the ( b − a ) G -invariant functions. If ( b − a ) G is ergodic,then this equals to (cid:90) X ∆ ah f dµ · (cid:90) X ∆ bh f dµ The limit of the average of this in absolute valuelim sup H →∞ E h ∈ Φ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) X ∆ ah f dµ · (cid:90) X ∆ bh f dµ (cid:12)(cid:12)(cid:12)(cid:12) is bounded by min { d a · (cid:107) f (cid:107) U · , d b · (cid:107) f (cid:107) U } and the claim follows by the van der corputlemma. If ( b − a ) G is not ergodic, then since ( b − a ) G is of index d b − a in G thereare at most d b − a ergodic components. In particular we can find a partition of X to( b − a ) G -invariant sets, X = (cid:83) d b − a i =1 A i such that P b − a is an integral operator with kernel (cid:80) d b − a i =1 A i ( x )1 A i ( y ). We conclude that (2.2) equals to, (cid:90) X (cid:90) X f ( x ) · f ( y ) · T ah f ( x ) · T bh f ( y ) d b − a (cid:88) i =1 A i ( x )1 A i ( y ) dµ ( x ) dµ ( y ) RGODIC AVERAGES AND KHINTCHINE RECURRENCE 11 Taking another average on h over any Følner sequence Ψ H , and apply the mean ergodictheorem for the action of T ah × T bh . The limit of the above becomes,(2.3) (cid:90) X (cid:90) X f ( x ) · f ( y ) d b − a (cid:88) i =1 H ( x, y )1 A i ( x )1 A i ( y ) dµ ( x ) dµ ( y )for some bounded T ah × T bh -invariant function H ( x, y ). It is classical that every T ah × T bh -invariant function can be written by sums of all products of d a eigenfunctions in x and d b eigenfunctions in y . Since 1 A i ( x ) is T ( b − a ) h -invariant it is also a sum of d b − a eigenfunctions.Let Z be the Kronecker factor, we conclude that the term in equation (2.3) is boundedby the minimum between d b − a · d a · max χ ∈ ˆ Z | (cid:104) f , χ (cid:105) | and d b − a · d b · max χ ∈ ˆ Z | (cid:104) f , χ (cid:105) | .Since the U -norm bounds the maximal Fourier coefficient the claim follows. To see thislet f ∈ L ( X ) be any function. We can decompose f with respect to the orthogonalprojection E ( ·| Z ) and write f = (cid:80) χ ∈ ˆ Z (cid:104) f, χ (cid:105) · χ + f (cid:48) , then (cid:107) f (cid:107) U = (cid:107) E ( f | Z ) (cid:107) U = (cid:88) χ ∈ ˆ Z | (cid:104) f, χ (cid:105) | ≥ max χ ∈ ˆ Z | (cid:104) f, χ (cid:105) | This clearly implies that (cid:107) f (cid:107) U ≥ max χ ∈ ˆ Z | (cid:104) f, χ (cid:105) | , by the van der Corput lemma wehave the promised inequality.It is left to show that the limit exists. By linearity we can reduce matters to the Kro-necker factor. For i = 1 , f i = E ( f i | Z ). Then, by approximating ˜ f , ˜ f by linearcombinations of eigenfunctions direct computation gives,lim N →∞ E g ∈ Φ N T ag ˜ f ( x ) · T bg ˜ f ( x ) = (cid:90) Z ˜ f ( xy a ) ˜ f ( xy b ) dµ Z ( y )in L . This completes the proof. (cid:3) Now, we generalize this for the tuple ( ag, bg, ( a + b ) g ). We have, Proposition 2.3 ( Z < ( X ) is characteristic for triple averages) . Let G be as in Theorem1.3 and let X be an ergodic G -system. Let f , f , f ∈ L ∞ ( X ) and for every i = 1 , , let ˜ f i = E ( f i | Z < ( X ) . Then, assuming that the following limits exists in L we have, lim N →∞ E g ∈ Φ N T ag f T bg f T ( a + b ) g f = lim N →∞ E g ∈ Φ N T ag ˜ f T bg ˜ f T ( a + b ) g ˜ f Proof. Let d a , d b , d b − a and d a + b denote the index of aG, bG, ( b − a ) G and ( a + b ) G in G respectively and let f , f , f ∈ L ∞ ( X ). By linearity it is enough to show that if either˜ f , ˜ f or ˜ f is zero then, lim N →∞ E g ∈ Φ N T ag f T bg f T ( a + b ) g f = 0By symmetry of the equation we can assume without loss of generality that ˜ f = 0.Moreover if we divide each function by a constant we can also assume that (cid:107) f (cid:107) ∞ , (cid:107) f (cid:107) ∞ and (cid:107) f (cid:107) ∞ are bounded by 1. Set x g = T ag f · T bg f · T ( a + b ) g f , then for every g, h ∈ G and N ∈ N we have, E g ∈ Φ N (cid:104) x g + h , x g (cid:105) = E g ∈ Φ N (cid:90) X T ag + ah f · T bg + bh f · T ( a + b )( g + h ) f · T ag f · T bg f · T ( a + b ) g f dµ since T ag is measure preserving the above equals to E g ∈ Φ N (cid:90) X ∆ ah f · T ( b − a ) g ∆ bh f · T bg ∆ ( a + b ) h f dµ By the previous lemma this average convergence in L . Observe that by the Cauchy-Schwartz inequality and since (cid:107) f (cid:107) ∞ ≤ 1, the absolute value of the above is smaller orequal to (cid:13)(cid:13) E g ∈ Φ N T ( b − a ) g ∆ bh f · T bg ∆ ( a + b ) h f (cid:13)(cid:13) L By the previous lemma, this is bounded by the square root of d a · (cid:13)(cid:13) ∆ ( a + b ) h f (cid:13)(cid:13) U ( X ) . Since (cid:107) · (cid:107) U ( X ) is a seminorm we conclude that for every Følner sequence Ψ H ,lim H →∞ E h ∈ Ψ H (cid:13)(cid:13) ∆ ( a + b ) h f (cid:13)(cid:13) U ( X ) ≤ d a + b · (cid:107) f (cid:107) U ( X ) Therefore, (cid:12)(cid:12)(cid:12) lim H →∞ E h ∈ Ψ H lim N →∞ E g ∈ Φ N (cid:104) x g + h , x g (cid:105) (cid:12)(cid:12)(cid:12) ≤ d a · d a + b · (cid:107) f (cid:107) U = 0and by the van der Corput lemma the claim follows. (cid:3) Generalized spectrum Let G be a countable abelian group, ( X, G ) an ergodic G -system and k ≥ 1. Roughlyspeaking, in this section we construct an extension ( Y, H ) such that every phase poly-nomial on X of degree < k has phase polynomial roots in Y .We begin with some definitions, Definition 3.1 (Phase polynomials) . Let X be an ergodic G -system, let k ≥ U be a compact abelian group. We say that a function P : X → U is a phase polynomialof degree < k if for every g , ..., g k we have that ∆ g ... ∆ g k P = 1 U . We let P Proposition 3.3. Let G be a countable abelian group. If m, k ≥ , X an ergodic G -system of order < k and P : X → S a phase polynomial of degree < m . Then, RGODIC AVERAGES AND KHINTCHINE RECURRENCE 13 • X is an abelian extension of Z Proposition 3.5 (Definition and properties of divisible groups) . A group ( H, · ) is saidto be divisible if for every h ∈ H and every ≤ n ∈ N there exists g ∈ H with g n = h .Divisible groups are injective in the category of discrete abelian groups. Namely, if H ≤ G are discrete abelian groups, and H is divisible then G ∼ = H ⊕ G/H . Given two abelian groups H and G , and an inclusion ı : H (cid:44) → G . We say that H isdivisible in G if for every n ∈ N and h ∈ H there exists g ∈ G with ı ( h ) = g n .We define the same for systems, Definition 3.6 (Divisible systems) . Let G be a countable abelian group and X bea G -system and k ≥ 2. We say that X is k -divisible if Spec ( X ) , ..., Spec k − ( X ) aredivisible. Similarly, if ( Y, H ) is an extension of X , then X is k -divisible in Y if for every1 ≤ i ≤ k − 1, Spec i ( X ) is divisible in Spec i ( Y ) with respect to the natural inclusion.In the ergodic case, if X is k -divisible then the group of phase polynomials of degree < k form a divisible group. More generally we have, Theorem 3.7 ( k -Divisible implies that P P/Q n is a phase polynomial of degree < d − 1. By induction hypothesis thereexists Q (cid:48) ∈ P Proposition 3.8 (Reducing C.L equations to the circle) . Let k ≥ and let X be anergodic k -divisible G -system. Let U be a compact abelian group and ρ : G × X → U bea cocycle. If for every χ ∈ ˆ U there exists a phase polynomial q χ : G × X → U of degree < k − and a measurable map F χ : X → U such that χ ◦ ρ = q χ · ∆ F χ . Then, thereexists a phase polynomial q : G × X → U and a measurable map F : X → U such that ρ = q · ∆ F .Proof. Let ρ : G × X → U be as in the proposition. Let K be the group of all pairs( χ, F ) for which the equation in the claim holds. Namely, K = { ( χ, F ) ∈ ˆ U × M ( X, S ) : ∃ c : G → S s.t. χ ◦ ρ = c · ∆ F }K is an abelian closed subgroup of ˆ U × M ( X, S ) and by the assumption the projection p : K → ˆ U is onto and ker p ∼ = P 1. It follows that χ (cid:55)→ q χ is also a homomorphismand so by pontryagin duality theorem there exists a measurable map F : X → U and aphase polynomial of degree < k − q : G × X → U such that F χ = χ ◦ F and χ ◦ q = q χ .Since the characters separates points we conclude that ρ = q · ∆ F , as required. (cid:3) Observe that every countable abelian group is a factor of a group with divisible dual(say Z ω ). Therefore for the sake of the proof of Theorem 1.18 it is enough to assumethat the group G has a divisible dual.Let k ≥ 1, then an element λ ∈ Spec k ( X ) is a multi-linear map. More generally wedefine, Definition 3.9. Let G be a countable abelian group, ( X, G ) a G -system and m ≥ 1. Wesay that λ : G m × X → S is a multi-cocycle if for every 1 ≤ i ≤ m , every g , ..., g m ∈ G and every g (cid:48) i ∈ G we have λ ( g , ..., g i · g (cid:48) i , ..., g m , x ) = λ ( g , ..., g i , ..., g m , x ) · λ ( g , ..., g (cid:48) i , ..., g m , T g x )In the special case where the multi-cocycle λ is constant in x , we say that λ is multi-linear .Moreover, we say that λ is symmetric if it is invariant under permutation of coordinates. RGODIC AVERAGES AND KHINTCHINE RECURRENCE 15 We denote by SML m ( G, S ) the symmetric multi-linear functions λ : G m → S and bySMC m ( G, X, S ) the symmetric multi-cocycles q : G m × X → S . Proposition 3.10. Let G be a countable abelian group such that ˆ G is divisible. Thenfor every m ≥ , SML m ( G, S ) is a divisible group.Proof. Let m ≥ λ ∈ SML m ( G, S ). If m = 1 then SML ( G, S ) ∼ = ˆ G and theclaim follows. We assume that m ≥ 2, then for every g , ..., g m choose an n ’th root for λ ( g , ..., g m ) in S and denote this root by µ ( g , ..., g m ). If we choose the same root forall permutations of coordinates we can also assume that µ ( g , ..., g m ) is symmetric.Fix g , ..., g m and let (cid:126)g = ( g , ...., g m ). Then for every h, h (cid:48) ∈ G we have that(3.3) k (cid:126)g ( h, h (cid:48) ) = µ ( hh (cid:48) , (cid:126)g ) µ ( h, (cid:126)g ) · µ ( h (cid:48) , (cid:126)g )is of order n . The map k (cid:126)g : G × G → C n defines a multiplication on the set B = G × C n by ( h, t ) · ( h (cid:48) , s ) = ( hh (cid:48) , k (cid:126)g ( h, h (cid:48) ) st ). Since k (cid:126)g is symmetric, B is abelian and we have ashort exact sequence 1 → C n → B → G → B = ˆ G ⊕ ˆ C n as discretegroups. Therefore, there is a cross section ˆ C n → ˆ B . By the pontryagin dual we obtaina cross section B → C n and the short-exact sequence above splits.We conclude by Proposition B.2 that there exists ν (cid:126)g : G → C n such that h (cid:55)→ µ ( h, (cid:126)g ) /ν (cid:126)g ( h ) is a homomorphism. Since µ is symmetric, and (cid:126)g is arbitrary, we canfind ν : G m +1 → C n such that µ/ν ∈ SML m ( G, S ). Since ν n = 1, we obtain that µ/ν isan n ’th root of λ , as required. (cid:3) We use the following result by Zimmer [28]. Definition 3.11 (Image and minimal cocycles) . Let G be a countable abelian group,let X be a G -system and let ρ : G × X → U be a cocycle into a compact subgroup. Theimage of ρ , U ρ is defined to be the closed subgroup generated by { ρ ( g, x ) : g ∈ G, x ∈ X } .We say that ρ is minimal if it is not ( G, X, U )-cohomologous to a cocycle σ with U σ (cid:8) U ρ Lemma 3.12. Let X be an ergodic G -system and ρ : G × X → U be a cocycle into acompact abelian group. Then, • ρ is ( G, X, U ) -cohomologous to a minimal cocycle. • X × ρ U is ergodic if and only if X is ergodic and ρ is minimal with image U ρ = U . We construct the extensions. Proposition 3.13. Let G be a countable abelian group with divisible dual and ( X, G ) be an ergodic G -system. Let d, m ∈ N be natural numbers and suppose that ( q n ) n ∈ N ∈ SM C m ( G, X, S ) are countably many phase polynomial of degree < d . Then there existsan ergodic extension π : ( Y, G ) → ( X, G ) and phase polynomials Q n : Y → S of degree < d + m with q n ( g , ..., g m , π ( y )) = ∆ g ... ∆ g m Q n ( y ) for all g , ..., g m ∈ G and µ Y -almostevery y ∈ Y . Proof. It is convenient to denote by q : G m × X → ( S ) N the multi-cocycle whose n ’thcoordinate is q n . We proceed by induction on m . For m = 1, q : G × X → ( S ) N is a co-cycle. Let τ n : G × X → ( S ) N be a minimal cocycle and F : X → ( S ) N with q = τ · ∆ F .We denote by V ≤ ( S ) N the image of τ and consider the extension Y = X × τ V . Let ı : V → ( S ) N be the embedding of V in ( S ) N , and let Q ( x, v ) = ı ( v ) · F ( x ). Then∆ g Q ( x, v ) = q ( g, x ), and so ∆ g Q n ( x, v ) = q n ( g, x ), where Q n is the n ’th coordinate of Q , as required. Let m ≥ m . For every g , ..., g m − the map g (cid:55)→ q ( g , ..., g m − , g, x ) isa cocycle. Therefore, as in the case m = 1, we can find an extension π : ˜ X → X and aphase polynomial Q : G m − × ˜ X → ( S ) N with q ( g , ..., g m − , g, π ( x )) = ∆ g Q g ,...,g m − ( x )for all g , ..., g m − , g ∈ G and a.e. x ∈ ˜ X .Since q is symmetric we can choose Q g ,...,g m − so that ( g , ..., g m − ) (cid:55)→ Q g ,...,g m − is in-variant under permutation of coordinates. Fix g , ..., g m − ∈ G and let (cid:126)g = ( g , ..., g m − ),we linearize Q g ,(cid:126)g in g . Observe that, K (cid:126)g = { ( g, Q ) ∈ G × P Theorem 3.14. Let G be a countable abelian group with divisible dual and k ≥ . Thenfor every ergodic system ( X, G ) there exists an extension ( Y, G ) , such that X is k -divisiblein Y . RGODIC AVERAGES AND KHINTCHINE RECURRENCE 17 Proof. Let X be as in the theorem. Fix k ∈ N , and let Spec( X ) = (cid:83) ki =1 Spec k ( X ).For every 1 ≤ i ≤ k , every λ ∈ Spec i ( X l ), and every n ∈ N choose an n ’th root λ n ∈ SM L i ( G, S ) for λ (Proposition 3.10). Then, as in the proposition above we canfind an extension Y such that { λ n : λ ∈ Spec( X ) , n ∈ N } belongs to Spec( Y ). Thiscompletes the proof. (cid:3) As a corollary we have, Theorem 3.15. Let G be a countable abelian group with divisible dual. Then everyergodic G -system X is a factor of a -divisible system.Proof. Let X be as in the theorem. Applying theorem 3.14 iteratively we obtain anincreasing sequence of extensions ( X n , G ) with the property that Spec( X n ) is divisible inSpec( X n +1 ). Let Y be the inverse limit of X n and recall that the factor map π : Y → X n induces a factor π : Z < ( Y ) → Z < ( X n ). It is classical that Z < ( Y ) is an inverse limit ofthe sequence ... → Z < ( X n ) → Z < ( X n − ) → ... → Z < ( X ) → Z < ( X )Let f be an eigenfunction of Y , then for every n ∈ N and g ∈ G we have T g E ( f | Z < ( X n )) = E ( T g f | Z < ( X n )) = λ g E ( f | Z < ( X n ))In particular, if E ( f | Z < ( X n )) (cid:54) = 0, then f is measurable with respect to Z < ( X n ).Therefore, for sufficiently large n , ∆ f ∈ Spec ( X n ). Since Spec ( X n ) is divisible inSpec ( Y ) this completes the proof. (cid:3) Divisible C.L systems are homogeneous We prove theorem 1.18. By Theorem 3.15 it is enough to show that, Theorem 4.1 (Divisible C.L systems are homogeneous) . Let G be a countable groupand let X be an ergodic -divisible ergodic G -system. Then the action of G ( Z < ( X )) on Z < ( X ) is transitive. We prove Theorem 4.1 and the properties mentioned in Remark 1.19. Proof. Let G be a countable abelian group and X be an ergodic G -system. By Proposi-tion A.11 we can write Z < ( X ) = Z < ( X ) × ρ U for some compact abelian group U anda cocycle ρ : G × Z < ( X ) → U . As usual we identify Z < ( X ) with a compact abeliangroup Z . Let χ ∈ ˆ U be a character and s ∈ Z , then by Proposition A.11 again, wecan find a character c s ( χ ) : G → S and a measurable map F s ( χ ) : Z → S such that∆ s ρ = c s ( χ ) · ∆ F s ( χ ). We conclude from Proposition 3.8 that for every s ∈ Z we can finda measurable map F s : Z → U such that S s,F s ∈ G ( Z < ( X ). Since the transformations S ,u for u ∈ S are also in G ( Z < ( X )) the action of this group on X is transitive.Let x = (1 , ∈ Z × U and let Γ be the stabilizer of x under the action of G ( X ). Then,Γ = { S ,F : F ∈ Hom( Z, U ) } is a totally disconnected closed subgroup of G ( Z < ( X )). By Theorem B.3 the projectionmap p : G ( Z < ( X )) → G ( Z < ( X )) / Γ is open and by Theorem B.4, Z < ( X ) is homeo-morphic to G ( Z < ( X )) / Γ. It follows that Z < ( X ) is isomorphic to G ( Z < ( X )) / Γ as a G -system where the action of g ∈ G on G ( Z < ( X )) / Γ is given by left multiplication by S g,ρ ( g, · ) . (cid:3) Simple homogeneous spaces. For completeness we show that any system witha nilpotent homogeneous structure as in Theorem 1.18 is an inverse limit of simplerhomogeneous spaces in which the stabilizer Γ is a discrete subgroup. We will not usethis result. Definition 4.2. Let G be a countable abelian group and let ( X, G ) be a C.L system.We say that X is a simple homogeneous space if the C.L group acts transitively on X and the stabilizer of any x ∈ X is a discrete subgroup. Proposition 4.3. Let G be a countable abelian group and let ( X, G ) be a C.L system.If G ( X ) acts transitively on X then X is an inverse limit of simple homogeneous spaces.Proof. Let X as in the proposition and write X = Z × ρ U where Z = Z < ( X ) is theKronecker factor. By Gleason-Yamabe theorem we can find a decreasing sequence ofclosed subgroups K n ≤ U such that (cid:84) n ∈ N K n = { } and the quotients L n = U/K n areLie groups. Let π n : U → L n be the projection map and let X n = Z × π n ◦ ρ L n . Since G ( X ) acts transitively on X , we have that for every s ∈ Z , there exists a measurablemap F : Z → U such that ∆ s ρ = c · ∆ F . Observe that if S s,F ∈ G ( X ) and ∆ s ρ = c · ∆ F for some c : G → U , then ∆ s π n ◦ ρ = π n ◦ c · ∆ π n ◦ F and S s,π n ◦ F ∈ G ( X n ). As S ,uK n belongs to G ( X n ) we conclude that the action on X n is transitive. Fix any x ∈ X n , thenthe stabilizer Γ n of x is homeomorphic to the discrete group hom( Z, U n ). Since X is aninverse limit of X n , the claim follows. (cid:3) The structure of a nilpotent system Let G be a countable abelian group and X be a C.L ergodic G -system such that theaction of G ( X ) on X is transitive. Write X = G ( X ) / Γ( X ) where Γ( X ) the stabilizer ofsome x ∈ X . We recall the definition of a group rotation. Definition 5.1. Let G be a countable abelian group. We say that a G -system X isa group rotation if it is isomorphic to a compact abelian group K and there exists ahomomorphism ϕ : G → K such T g k = ϕ ( g ) k for every g ∈ G and k ∈ K .It is well known that the Kronecker factor is the maximal group rotation. Namely, Theorem 5.2 (Maximal property of the Kronecker factor) . Let G be a countable abeliangroup and X be a G -system. Then any group rotation factor Y of X is a factor of Z < ( X ) . RGODIC AVERAGES AND KHINTCHINE RECURRENCE 19 Recall that any C.L system can be written as X = Z × ρ U where Z is the Kroneckerfactor, U a compact abelian group and ρ : G × Z → U is a cocycle (Proposition A.11). Inthe case where the action of G ( X ) on X is transitive, it is possible to obtain the groups Z and U from the homogeneous group and the stabilizer. Namely, Lemma 5.3. Let G be a countable abelian group and X = Z × ρ U be an ergodic C.L G -system where Z is the Kronecker factor and suppose that the action of G ( X ) on X is transitive. If G is an open subgroup of G ( X ) which contains the embedding of G in G ( X ) , then Z ∼ = G / G Γ and U ∼ = G where Γ := Γ( X ) ∩ G and G is the closed subgroupgenerated by the commutators { [ a, b ] : a, b ∈ G} .Proof. First we prove that G / Γ ∼ = G ( X ) / Γ( X ) as measure spaces. To see this observe thatthe projection p : G ( X ) → G ( X ) / Γ( X ) is an open map (Theorem B.3). Therefore, p ( G )is a G invariant open (and closed) subset of G ( X ) / Γ( X ), hence by ergodicity p ( G ) = X .We conclude that the map g Γ (cid:55)→ g Γ( X ) from G / Γ to G ( X ) / Γ( X ) is an isomorphism.In particular, there exists a factor map π : G / Γ → Z . Direct computation shows that G acts trivially on Z and π factors through G . By Lemma 5.2, π : G / G Γ → Z is anisomorphism, hence Z ∼ = G / G Γ.Let p : G ( X ) → Z be the projection map S s,F (cid:55)→ s . The group p ( G ) is an open andclosed G -invariant subgroup of Z and so by ergodicity p ( G ) = Z . Choose a cross-section s (cid:55)→ S s,F s as in Theorem B.3. We have,[ S g,σ ( g ) , S s,F s ] = S , ∆ sσ ∆ Fs and ∆ s σ ∆ g F s is a constant in U . We identify G with the closed subgroup generated bythese constants. Suppose by contradiction that G (cid:8) U , then there exists a non-trivialcharacter χ : U → S such that ∆ s χ ◦ σ = ∆ χ ◦ F s . Theorem A.5 implies that factor Y = Z × χ ◦ σ χ ( U ) is isomorphic to a group rotation and Theorem 5.2 provides a contradiction. (cid:3) We need the following weaker notion of divisibility. Definition 5.4. Let U be an abelian group and n ∈ N . We denote by U n := { u n : u ∈ U } and say that U is n -divisible if U n = U .As a corollary of the previous lemma we conclude, Corollary 5.5. Let G be a countable abelian group and a, b ∈ Z as in Theorem 1.3. Let ( X, G ) be an ergodic C.L system and suppose that G ( X ) acts transitively on X . Thenthe commutator subgroup G ( X ) is a , b and ( a + b ) -divisible.Proof. By the previous lemma, we can write X = Z × σ U where U = G ( X ) . Fix anumber m ∈ N and suppose by contradiction that U is not m -divisible. Then thereexists a non-trivial character χ : U → C m . Let s (cid:55)→ S s,F s be a cross-section from Z to G ( X ) and let c s : G → S such that ∆ s χ ◦ ρ = c s · ∆ F s . where [ a, b ] = a − b − ab as usual. Observe that c ms ∈ B ( G, X, S ) is an eigenvalue. If in addition m is such that mG isof finite index in G , then the set { c s : s ∈ Z } is at most countable. Thus, the group Z (cid:48) = { s ∈ Z : ∆ s χ ◦ ρ } is a G -invariant open subgroup of Z and by ergodicity, Z (cid:48) = Z .As before, Theorem A.5 implies that the extension by χ ◦ ρ is a group rotation andTheorem 5.2 provides a contradiction. (cid:3) Remark 5.6. In the previous corollary, since at least one of a, b, a + b is even the group G ( X ) is automatically 2-divisible.6. Limit formula and point-wise convergence We prove the following point-wise convergence for some multiple ergodic averages ona 2-step homogeneous space where the homogeneous group is the C.L group. Theorem 6.1 (Limit formula) . Let X = G / Γ be an ergodic C.L G -system, where G isthe C.L group and suppose that G is -divisible. Let µ G denote the Haar measure on G ,then for every k ∈ N , every f , f , ..., f k ∈ L ∞ ( X ) and µ -almost every x ∈ X we have, lim N →∞ E g ∈ Φ N k (cid:89) i =1 T ig f i ( x ) = (cid:90) G / Γ (cid:90) G k (cid:89) i =1 f i ( xy i y ( i ) ) dµ G ( y ) dµ ( y )(6.1) with the abuse of notation that f ( x ) = f ( x Γ) . As a corollary we get, Corollary 6.2. Let a, b ∈ Z . In the settings of Theorem 6.1, choose k = a + b , let h , h , h ∈ L ∞ ( X ) be any bounded functions and set f a = h , f b = h , f a + b = h and f i = 1 for all i (cid:54) = a, b, a + b . Then, for µ -almost every x ∈ X we have, lim N →∞ E g ∈ Φ N T ag h ( x ) T bg h ( x ) T ( a + b ) g h ( x ) = (cid:90) G / Γ (cid:90) G h ( xy a y ( a ) ) h ( xy b y ( b ) ) h ( xy a + b y ( a + b ) ) dµ G ( y ) dµ ( y )(6.2)We follow an argument from Bergelson Host and Kra [4]. Let, ı : G × G × G → G k +1 ı ( g, g , g ) = ( g, gg , gg g , ..., gg k g ( k ) )We denote by ˜ G the image of ı . In [19] Leibman proved that ˜ G is a 2-step nilpotentgroup. The subgroup ˜Γ = ı (Γ × Γ × { e } ) is a closed subgroup of ˜ G and the quotientspace ˜ X = ˜ G / ˜Γ is compact. Let ˜ µ be the Haar measure on this space, and define an actionof G × G on ( ˜ X, ˜ µ ) by left multiplication with g (cid:52) := ( g, g, g, g ) and g (cid:63) = (1 , g, g , ..., g k )where g is identified with the measure-preserving transformation T g : X → X in G ( X ). RGODIC AVERAGES AND KHINTCHINE RECURRENCE 21 In Lemma 6.6 below we prove that this action is uniquely ergodic. Assume this for now,and fix x ∈ X . We consider the compact polish space˜ X x := { ( x , x , .., x k ) ∈ X k : ( x, x , x , ..., x k ) ∈ ˜ X } Bergelson Host and Kra showed that the group ˜ G (cid:63) = { ( g , g g , ..., g k g ( k ) ) : g ∈ G , g ∈G } acts transitively on this space and ˜ X x ∼ = ˜ G (cid:63) / ˜Γ (cid:63) where ˜Γ (cid:63) = { ( γ, γ , ..., γ k ) : γ ∈ Γ } .Observe that since ı is injective, it induces an isomorphism of G -systems, ˜ ı : G / Γ × G → ˜ X x where the action of G on G × G is given by T g ( y , y ) = ( g [ g, x ] y , [ g, y ] y ).We continue assuming that the action of G × G on ˜ X is uniquely ergodic. Let ˜ µ x be theHaar measure on ˜ X x , Bergelson Host and Kra [4] proved, Lemma 6.3. ˜ µ = (cid:90) X δ x ⊗ ˜ µ x dµ ( x )We can now prove Theorem 6.1. Proof. Since continuous functions are dense in L ∞ ( X ), it is enough to prove the theoremfor continuous f , f , ..., f k . Let F : G / Γ k → C , F ( x , x , ..., x k ) = f ( x ) · f ( x ) · ... · f k ( x k ),we can write average (6.1) as E g ∈ Φ N ( T g × T g × ... × T kg ) F ( x, x, ..., x )Recall that every element in the orbit of ( x, x, ..., x ) under the transformation T g × T g × ... × T kg belongs to ˜ X x . Thus, by the mean ergodic theorem average (6.1) convergespointwise everywhere to a function φ ( x ) on X . Let f be any continuous function on X .We have, (cid:90) f ( x ) φ ( x ) dµ ( x ) = lim N →∞ (cid:90) E g ∈ Φ N f ( x ) · k (cid:89) i =1 f i ( T ig x ) dµ ( x )Since µ is G -invariant, the above equals tolim N →∞ (cid:90) E g,h ∈ Φ N f ( T h x ) k (cid:89) i =1 f i ( T ig + h x ) dµ ( x )since ( x, x, ..., x ) belongs to ˜ G / ˜Γ and the action of G × G by h (cid:52) and g (cid:63) is uniquely ergodicwe conclude by the mean ergodic theorem that this converges everywhere to (cid:90) ˜ X f ( x ) k (cid:89) i =1 f ( x i ) d ˜ µ ( x , x , ..., x k )which by Lemma 6.3 is equals to, (cid:90) X f ( x ) (cid:32)(cid:90) ˜ X x k (cid:89) i =1 f i ( x i ) d ˜ µ x ( x , ..., x k ) (cid:33) dµ ( x ) As this holds for every continuous function f , we conclude that φ ( x ) = (cid:90) ˜ X x k (cid:89) i =1 f i ( x i ) d ˜ µ x ( x , ..., x k ) = (cid:90) G / Γ (cid:90) G f ( xy ) f ( xy y ) · ... · f ( xy k y ( k ) ) dµ G ( y ) dµ ( y )as required. (cid:3) By Parry [23] an ergodic action on ˜ G/ ˜Γ is uniquely ergodic. Therefore, in order tocomplete the proof of Theorem 6.1 it is left to prove that the action of G × G is ergodic.We need a computation first, Lemma 6.4. Let ˜ G as in the proof of Theorem 6.1. If V ≤ ˜ G is an open subgroup whichcontains g (cid:52) and g (cid:63) for all g ∈ G then, V = { ( g, gg , gg g , ..., gg k g ( k ) ) : g, g , g ∈ G } Proof. Let ı : G × G × G → ˜ G as in the proof of Theorem 6.1. Let L , L (cid:48) ≤ G be opensubgroups such that L × L (cid:48) × { e } ≤ i − ( V ). Since g (cid:52) ∈ V we can assume that g ∈ L and since g (cid:63) ∈ V that g ∈ L (cid:48) . By taking an intersection we may assume that L = L (cid:48) ,and by Lemma 5.3 we have that L = G . Let g, g , g ∈ G ,For every s , s ∈ L , ( s , s , ..., s ) and ( s , s , ..., s ) belong to V and therefore ( g, g, ..., g ) ∈ V .For every t , t ∈ L , we have that ( t , t , ..., t ) and ( e, t , t , ..., t k ) belong to V . Sincethe commutator is a bilinear map we conclude that ( e, g , g , ..., g k ) ∈ V .Finally, for every r , r ∈ L , ( e, r , r , ..., r k ) and ( e, r , r , ..., r k ) belong to V and( e, [ r , r ] , [ r , r ] , ..., [ r , r ] k ) ∈ V Since ( r , r , ...., r ) also belongs to V , ( e, [ r , r ] , [ r , r ] , ..., [ r , r ] k ) ∈ V . We concludethat ( e, [ r , r ] − , [ r , r ] − , ...., [ r , r ] k − k ) ∈ V and since G is 2-divisible, ( e, e, g , g ( ) , ..., g ( k ) ) ∈ V . Combining everything we see that V = { ( g, gg , gg g , ..., gg k g ( k ) ) : g, g , g ∈ G } as required. (cid:3) As a corollary we have, Corollary 6.5. The induced action of g (cid:52) and g (cid:63) on ˜ G / ˜ G Γ is ergodic.Proof. The map ı induces a factor map G / G Γ × G / G Γ → ˜ G / ˜ G Γ. The lift of g (cid:52) and g (cid:63) in G / G Γ × G / G corresponds to T g × T g and Id × T g respectively. Since G / Γ is ergodicthe claim follows. (cid:3) We finally have, Lemma 6.6. The action of G × G on ˜ G / ˜Γ by g (cid:52) and g (cid:63) is ergodic. RGODIC AVERAGES AND KHINTCHINE RECURRENCE 23 Proof. We follow an argument of Parry [24]. Let f : ˜ G / ˜Γ → S be an invariant function.The compact abelian group ˜ G acts on L ( ˜ G / ˜Γ). Therefore we can find eigenfunctions f λ , such that f = (cid:80) λ a λ f λ where a λ ∈ C and λ is a character of ˜ G . By the uniqueness ofthe decomposition, it follows that f λ is also an eigenfunction with respect to the actionof g (cid:52) and g (cid:63) . By Corollary 6.5 we can assume that f λ takes values in S . Fix u ∈ ˜ G ,and let h = g (cid:52) or h = g (cid:63) . Then, f λ ( uhx ) = f λ ([ u − , h − ] hux ) = λ ([ u − , h − ]) f λ ( hux ) = λ ([ u − , h − ]) c h f λ ( ux )for some constant c h ∈ S . Therefore, the function ∆ u f λ ( x ) is an eigenfunction withrespect to the action of G × G and is invariant under the action of ˜ G . By Corollary 6.5and Lemma 3.2 the set { ∆ u f λ : u ∈ ˜ G} is countable modulo constants. It follows that V λ := { u ∈ ˜ G : ∆ u f λ is a constant } is an open subgroup. Observe that u (cid:55)→ ∆ u f λ is ahomomorphism from V λ → S and is therefore trivial on the commutator subgroup ( V λ ) which by Lemma 6.4, equals to ˜ G . We conclude that f is invariant under the action of˜ G , and by Corollary 6.5 is a constant. (cid:3) Proof of the Khintchine type recurrence In this section we finish the proof of the Khintchine type recurrence. First we prove alifting lemma which allow us to replace any system ( X, G ) with an extension ( Y, H ). Lemma 7.1. Let G be a countable abelian group and ( X, T g ) be a G -system. Let ϕ : H → G be a surjective homomorphism and ( Y, S h ) be an H -extension of X with a factormap π : Y → X . Then for any Følner sequence Φ N of G there exists a Følner sequence Ψ N of H such that for every N ∈ N and f , f , f ∈ L ∞ ( X ) , E h ∈ Ψ N S ah f ◦ π · S bh f ◦ π · S ( a + b ) h f ◦ π = φ ◦ π where, φ = E g ∈ Φ N T ag f · T bg f · T ( a + b ) g f Proof. For each g ∈ Φ N choose a single representative h g ∈ H and let Φ (cid:48) N = { h g : g ∈ Φ N } . Let K = ker ϕ and choose a Følner sequence Φ KN for K . Then, Ψ N = Φ (cid:48) N · Φ KN isa Følner sequence for H with ϕ (Ψ N ) = Φ N . Since π ◦ S h = T ϕ ( h ) ◦ π for all h ∈ H theclaim follows. (cid:3) The rest of the proof follows an argument of Frantzikinakis [12]. Proof. Let ( X, B , µ, G ) be an ergodic G -system and let 0 (cid:54) = a, b ∈ Z as in Theorem 1.3.We first prove the theorem in the case where a and b are coprime.For every f ∈ L ∞ ( X ) we denote ˜ f = E ( f | Z < ( X )). Recall that the Kronecker factor isa group rotation, and denote by α g ∈ Z < ( X ) the rotation defined by g ∈ G . Then, Claim: For every continuous function η : X → R + with η = ˜ η and every f , f , f ∈ L ∞ ( X ) we havelim N →∞ E g ∈ Φ N η ( α g ) T ag f · T bg f · T ( a + b ) g f = lim N →∞ E g ∈ Φ N η ( α g ) T ag ˜ f · T bg ˜ f · T ( a + b ) g ˜ f Proof. Approximating η by linear combinations of eigenfunctions, we see that it is enoughto prove the claim in the case where η is a character. Choose s, t ∈ Z such that η sa · η tb = η ,and apply Proposition 2.3 for η s · f , η t · f and f . (cid:3) Assume by contradiction that Theorem 1.3 fails. Then one can find ε > N for G such that(7.1) µ ( A ∩ T ag A ∩ T bg A ∩ T ( a + b ) g A ) < µ ( A ) − ε for every g ∈ (cid:83) N Φ N .By Theorem 1.18, we can find a surjective homomorphism ϕ : H → G and an H -extension ( ˜ X, H ) of ( X, G ) such that the factor Y = Z < ( ˜ X ) is a C.L system and Y = G ( Y ) / Γ. Note that since every extension in the proof of Theorem 1.18 only extendsthe Kronecker factor of X we have by Lemma 5.3 that G ( Y ) = G ( Z < ( X )) .Let f ∈ L ∞ ( X ), we can push-forward ˜ f to a function on Z < ( X ) and then let f (cid:63) denote the pullback of this function to the Y . Let Φ HN be any Følner sequence for H . ByCorollary 6.2 the average E h ∈ Φ HN η ( β h ) S hg f (cid:63) ( y ) S bh f (cid:63) ( y ) T ( a + b ) g f (cid:63) ( y )converges to (cid:90) Y (cid:90) G ( Y ) η (cid:63) ( y ) f (cid:63) ( yy a y ( a ) ) f (cid:63) ( yy b y ( b ) ) f (cid:63) ( yy a + b y ( a + b ) ) dµ G ( Y ) ( y ) dµ Y ( y )where β h corresponds to the action of h ∈ H on the Kronecker factor of Y .Let Φ HN be the Følner sequence from lemma 7.1 and f = f = f = 1 we conclude,(7.2) lim N →∞ E g ∈ Φ N η ( α g ) = lim N →∞ E h ∈ Φ HN η (cid:63) ( β h ) = 1Now let η be arbitrary and set f = f = f = f = f = 1 A , we conclude that the average E g ∈ Φ HN η (cid:63) ( β h ) (cid:90) Y f (cid:63) ( y ) · S hg f (cid:63) ( y ) · S bg f (cid:63) ( y ) · S ( a + b ) h f (cid:63) ( y ) dµ Y ( y )converges to (cid:90) Y (cid:90) Y (cid:90) G ( Y ) η ( y ) f (cid:63) ( y ) f (cid:63) ( yy a y ( a ) ) f (cid:63) ( yy b y ( b ) ) f (cid:63) ( yy a + b y ( a + b ) ) dµ G ( Y ) ( y ) dµ Y ( y ) dµ Y ( y )Since η is an arbitrary continuous function, we can approximate the indicator functions µ ( B ( G ( Y ) ,δ ) · B ( G ( Y ) , δ ) where B ( G ( Y ) , δ ) denote the ball of radius δ around all elementsof G ( Y ) for some δ > 0. Since translations are continuous in L , taking a limit as δ → (cid:90) Y (cid:90) G ( Y ) ×G ( Y ) f (cid:63) ( y ) f (cid:63) ( yy a y ( a ) ) f (cid:63) ( yy b y ( b ) ) f (cid:63) ( yy a + b y ( a + b ) ) dµ G ( Y ) ×G ( Y ) ( y , y ) dµ Y ( y ) RGODIC AVERAGES AND KHINTCHINE RECURRENCE 25 we integrate everything to get this equals to (cid:90) Y (cid:90) G ( Y ) f (cid:63) ( yy (cid:48) ) f (cid:63) ( yy (cid:48) y a y ( a ) ) f (cid:63) ( yy (cid:48) y b y ( b ) ) f (cid:63) ( yy (cid:48) y a + b y ( a + b ) ) dµ G ( Y ) ( y (cid:48) , y , y ) dµ Y ( y )By proposition B.5 we can write the above integral as (cid:90) Y (cid:90) G ( Y ) f (cid:63) ( u a + b y ) f (cid:63) ( t · u b − a y ) f (cid:63) ( t · v b − a y ) f (cid:63) ( v a + b y ) dµ G ( Y ) ( t, u, v ) dµ Y ( y )This clearly equals to (cid:90) Y (cid:90) G ( Y ) (cid:18)(cid:90) G ( Y ) f (cid:63) ( u a + b y ) f (cid:63) ( tu b − a y ) dµ G ( Y ) ( u ) (cid:19) dµ G ( Y ) ( t ) dµ Y ( y )We take the square outside and change variables, the above is greater or equal to (cid:90) Y (cid:90) G ( Y ) (cid:0) f (cid:63) ( ty ) dm G ( Y ) ( t ) (cid:1) dµ Y ( y ) = (cid:18)(cid:90) Y f (cid:63) ( x ) dµ y ( y ) (cid:19) = µ ( A ) We conclude by Lemma 7.1 that for every ε > 0, for sufficiently large N and a suitable η we have,(7.3) E g ∈ Φ N η ( a g ) µ ( A ∩ T ag A ∩ T bg A ∩ T ( a + b ) g A ) > µ ( A ) − ε/ a and b are co-prime, equations (7.2) and (7.3) contradict equation (7.1) andthe claim follows.Now let a and b be arbitrary non-zero integers and write a = a (cid:48) d, b = b (cid:48) d where a (cid:48) and b (cid:48) are coprime. Since aG and bG are of finite index in G so is dG and so X has finitelymany ergodic components with respect fo dG with the same Kronecker factor. Choose η as before (the same η for all ergodic components) and let µ = k (cid:80) ki =1 µ i . Since a (cid:48) , b (cid:48) are coprime by equation (7.3) we have E g ∈ Φ N η ( α g ) µ i ( A ∩ T a (cid:48) g A ∩ T b (cid:48) g A ∩ T ( a (cid:48) + b (cid:48) ) g A ) > µ i ( A ) − ε/ ≤ i ≤ k . Since E ≤ i ≤ k ( µ i ( A ) ) ≥ µ ( A ) , we conclude as before that the set { g ∈ dG : µ ( A ∩ T a (cid:48) g ∩ T b (cid:48) g A ∩ T ( a (cid:48) + b (cid:48) ) g A ) > µ ( A ) − ε } is syndetic. Since dG is of finite index in G this is equivalent to the claim in thetheorem. (cid:3) Appendix A. Abelian extensions and phase polynomials In this section we summarize previous results related to abelian extensions and phasepolynomials.The following proposition were proved by Host and Kra for Z -actions [18]. The sameargument holds for all countable abelian groups (for details see [1]). Proposition A.1. Let k ≥ , G be a countable abelian group and X be an ergodic G -system. Then Z Definition A.2 (Cubic measure spaces) . [18, Section 3] Let G be a countable abeliangroup and X = ( X, B , µ, G ) be a G -system. For each k ≥ X [ k ] =( X [ k ] , B [ k ] , µ [ k ] , G [ k ] ) where X [ k ] = X k is the product of 2 k copies of X , B [ k ] = B k and G [ k ] = G k acting on X [ k ] in the obvious manner. We define the cubic measures µ [ k ] and σ -algebras I k ⊆ B [ k ] inductively. I is defined to be the σ -algebra of invariant sets in X , and µ [0] := µ . Once µ [ k ] and I k are defined, we identify X [ k +1] with X [ k ] × X [ k ] anddefine µ [ k +1] by the formula (cid:90) f ( x ) f ( y ) dµ [ k +1] ( x, y ) = (cid:90) E ( f |I k )( x ) E ( f |I k )( x ) dµ [ k ] ( x )For f , f functions on X [ k ] and E ( ·|I k ) the conditional expectation, and I k +1 being the σ -algebra of invariant sets in X [ k +1] .This leads to the following generalization of Definition 1.17. Definition A.3 (The Host-Kra group for a system of order < k .) . Let G be a countableabelian group and k ≥ 1. We define G ( X ) to be the group of measure preservingtransformations t : X → X which satisfies the following property: For every l > 0, thetransformation t [ l ] : X [ l ] → X [ l ] , t [ l ] ( x ω ) ω ∈ k = ( tx ω ) ω ∈ k , leaves the measure µ [ l ] invariantand acts trivially on the invariant σ -algebra I l .Equipped with the topology of convergence in measure G ( X ) is a ( k − Definition A.4 (Functions of type < k ) . Let G be a countable abelian group, let X = ( X, B , µ, G ) be a G -system. Let k ≥ X [ k ] be the cubic system associatedwith X . • For each measurable f : X → U , we define a measurable map d [ k ] f : X [ k ] → U to be the function d [ k ] f (( x w ) w ∈{− , } k ) := (cid:89) w ∈{− , } k f ( x w ) sgn( w ) where sgn( w ) = w · w · ... · w k • Similarly for each measurable ρ : G × X → U we define a measurable map d [ k ] ρ : G × X [ k ] → U to be the function d [ k ] ρ ( g, ( x w ) w ∈{− , } k ) := (cid:89) w ∈{− , } k ρ ( g, x w ) sgn( w ) RGODIC AVERAGES AND KHINTCHINE RECURRENCE 27 • A function ρ : G × X → U is said to be a function of type < k if d [ k ] ρ is a( G, X [ k ] , U )-coboundary.We have, Theorem A.5. Let k, m ≥ and let G be a countable abelian group. Let ( X, G ) is anergodic G -system of order < k and ρ : G × X → U is a cocycle into some compact abeliangroup U . Then, • X × ρ U is of order < k + 1 if and only if ρ is of type < k . • If ρ is of type < k − then X × ρ U if of type < k .Proof. The first claim is proved in [18, Proposition 6.4] and the second in [18, Proposition7.6] for Z -actions. The general case follows by the same argument. (cid:3) In particular this implies that the C.L factor of an ergodic G -system is an abelianextension of the Kronecker factor by a cocycle of type < 2. The following definition isclosely related to the Conze-Lesigne equations in Definition 1.13. Definition A.6 (Automorphism) . Let X be a G -system. A measure-preserving trans-formation u : X → X is called an automorphism if the induced action on L ( X ) by V u ( f ) = f ◦ u commutes with the action of G . In this case we define the multiplicativederivative with respect to u by ∆ u f = V u f · f .The following result is due to Bergelson Tao and Ziegler [6, Lemma 5.3] Lemma A.7 (Differentiation by an automorphism decreases the type) . Let k, m ≥ , G be a countable abelian group, X be a G -system and ρ : G × X → S be a cocycle of type < m . Then for every automorphism t : X → X which preserves Z Lemma A.8. In the settings of Lemma A.7 if f is a phase polynomial of degree < m then ∆ t f ( x ) is of degree < m − min( m, k ) .Proof. [6, Lemma 8.8]. (cid:3) The following characterization of phase polynomials of degree < k is due to BergelsonTao and Ziegler [6]. Lemma A.9. Let G be a countable abelian group and X be an ergodic G -system. Thena function f : X → S is a phase polynomial of degree < k if and only if d [ k ] f ( x ) = 1 for µ [ k ] -almost every x ∈ X [ k ] . It is natural to ask whether a cocycle of type < k is cohomologous to a phase polyno-mial of degree < k . This is true for F ωp -systems [6] but wrong in general (see e.g. [17] or[22, Section 9]). However in the case k = 1 we have, Theorem A.10. [21] Let G be a countable abelian group. Let X be an ergodic G -systemand ρ : G × X → S be a cocycle of type < . Then, there exists a character c : G → S and a measurable map F : X → S such that ρ ( g, x ) = c ( g ) · ∆ g F ( x ) , for every g ∈ G and µ -almost every x ∈ X . We conclude, Proposition A.11. Let G be a countable abelian group. If m, k ≥ , X is an ergodic G -system of order < k and P : X → S a phase polynomial of degree < m . Then, • There exists a compact abelian group U and a cocycle ρ : G × X → U such that Z 2. Therefore,by Lemma A.7, ∆ s ρ is of type < s ∈ Z and the C.L equation follows byTheorem A.10.Let P : X → S be as in the theorem, then by Lemma A.8 we have that ∆ u P is of degree < m − min { m, k } . In particular, if k ≥ m then P is invariant under U . Applying thisargument iteratively (by induction) gives the desired result. The last claim is a directapplication of Lemma A.9 and Theorem A.1. (cid:3) Appendix B. Results about topological groups and a computation B.1. Divisible and injective groups. We begin with some definitions, Definition B.1. Let Z and U be locally compact abelian groups. A function k : Z × Z → U is called a cocycle if for every r, s, t ∈ Z we have,(B.1) k ( rs, t ) · k ( r, s ) = k ( r, st ) · k ( s, t )Moreover, a cocycle is symmetric if(B.2) k ( s, t ) = k ( t, s )for every s, t ∈ Z Proposition B.2. Let Z and U be locally compact abelian groups and let k : Z × Z → U be a symmetric cocycle. If one of the following holds • U is a torus. Or, • U, Z are discrete and U is divisible.then there exists a continuous function ϕ : Z → U such that k ( s, t ) = ϕ ( st ) ϕ ( s ) ϕ ( t ) . RGODIC AVERAGES AND KHINTCHINE RECURRENCE 29 Proof. Without loss of generality we may assume that k (1 , 1) = 1 U . From equation (B.1)we see that k (1 , t ) = k ( t, 1) = 1 U for all t ∈ Z . The cocycle k induces a multiplicationon the set K = Z × U by, ( s, u ) · ( t, v ) = ( st, k ( s, t ) uv )Equations (B.1) and (B.2) imply that K is an abelian group. Observe, that we have ashort exact sequence 1 → U ι → K p → Z → ι ( u ) = (1 , u ) and p ( z, u ) = z . By the assumptions in the claim the short exactsequence splits. Therefore, there exists an homomorphism q : Z → K with p ( q ( z )) = z . Let ϕ : Z → U be such that q ( z ) = ( z, ϕ ( z )). Since q is a homomor-phism, the claim follows. (cid:3) B.2. Polish spaces and group actions. Polish groups, and polish spaces (homoge-neous spaces in particular) play an important role in this paper.Below we summarize important results. Theorem B.3 (The open mapping Theorem) . [5] Let G and H be Polish groups and let p : G → H be a surjective continuous homomorphism. Then p is open and there exists aBorel cross-section s : H → G such that p ◦ s = Id . This theorem leads to the following result about quotient spaces [11]. Theorem B.4. If G is a locally compact polish group which acts transitively on a compactmetric space X . Then for any x ∈ X the stabilizer Γ = { g ∈ G : gx = x } is a closed and X is homeomorphic to G / Γ . B.3. A computation. We will need the following computation for the Khintchine re-currence. Proposition B.5. Let a, b ∈ Z be coprime and U be a compact abelian group. Supposethat U is a, b, a + b and b − a divisible then the sets A = { ( g, gg a g ( a ) , gg b g ( b ) , gg a + b g ( a + b ) ) : g, g , g ∈ U } and B = { ( u a + b , t · u b − a , tv b − a , v a + b ) ∈ U : u, t, v ∈ U } are equal.Proof. We first prove that A ⊆ B . To see this fix any g, g , g ∈ U . Let s ∈ U be such that s = g , choose u ∈ U such that u a + b = g and set v = ug s a + b − and t = gg a g ( a ) · u a − b .Clearly, v a + b = gg a + b g ( a + b ) and it left to show that gg a g ( a ) · u a − b · v b − a = gg b g ( b ) We substitute v = ug s a + b − we get g b − a s ( a + b − b − a ) = g b − a g ( b ) − ( a ) since either ( a + b − 1) or b − a is even we get that the equality holds.As for the second inclusion fix any u, t, v ∈ U . Let g = u a + b , for every g ∈ U choose s = s ( g ) such that s = g and set g = vu − s − a − b . It is left to find g such that thefollowing equations hold (cid:40) u a + b · ( vu − s − a − b ) a s a − a = t · u b − a u a + b · ( vu − s − a − b ) b s b − b = t · v b − a Rearranging the equations we get, (cid:110) s ab = t − · ( uv ) a Since U is a and b divisible it is also ab divisible and so an s like that exists and we cantake g = s . (cid:3) References [1] E. Ackelsberg, V. Bergelson, A. 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