An uncountable ergodic Roth theorem and applications
Polona Durcik, Rachel Greenfeld, Annina Iseli, Asgar Jamneshan, José Madrid
aa r X i v : . [ m a t h . D S ] J a n UNIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES FOR TWOCOMMUTING ACTIONS OF AMENABLE GROUPS
POLONA DURCIK, RACHEL GREENFELD, ANNINA ISELI, ASGAR JAMNESHAN, AND JOSÉ MADRIDA bstract . We establish a multiple recurrence theorem for two commuting actions of a discrete(not necessarily countable) amenable group on a (not necessarily separable) probability algebra.Based on it, we further establish that for all ε >
0, all arbitrary uniformly amenable sets G of groups, all pairs of commuting measure-preserving actions of a group in G on an arbitraryprobability algebra ( X , µ ), and all E ∈ X with µ ( E ) ≥ ε , the set of multiple recurrence times of E is syndetic in a uniform fashion over all such measure-preserving dynamical systems, wherethe degree of uniform syndeticity depends only on ε and the function F measuring the uniformamenability of G . New ingredients in the proofs of our two main results are recently developedbasic results in uncountable measure theory and ergodic theory, the use of conditional analysistechniques, and an ultralimit analysis of Banach densities. We also deduce from our multiplerecurrence theorem that the set of triangular configurations in a dense subset of Γ × Γ is syndeticfor any discrete amenable group Γ .
1. I ntroduction
The subject of ergodic Ramsey theory was pioneered by Furstenberg in [20], with an ergodictheoretic proof of Szemerédi’s deep theorem [41] on the existence of arbitrarily long arithmeticprogressions in sets of integers of positive density. The ergodic theoretic method of Fursten-berg’s proof has proven to have wide implications in density Ramsey theory by leading to sev-eral significant extensions of Szemerédi’s theorem. Our focus in this work is on Furstenberg’sergodic Roth theorem [20, Theorem 3.5], which is the case of linear 3-term progressions in Sze-merédi’s theorem established originally by Roth [38], and its extensions. One of the significantextensions of the latter theorem is the ergodic Roth theorem for countable amenable groups byBergelson, McCutcheon and Zhang [7]. In this paper we manage to extend the main results ofthe latter work, establishing an uncountable and a uniform version of [7, Theorem 5.2] and anuncountable version of its main combinatorial application [7, Theorem 6.1]. Before formallystating our results, we review some further related results from the literature to locate ours in aproper context.Recall that Furstenberg’s multiple recurrence theorem states that if T : X → X is a measure-preserving transformation of a probability space ( X , X , µ ), then for every E ∈ X with µ ( E ) > Date : February 1, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Uncountable ergodic theory, ergodic Ramsey theory, ergodic Roth theorem, amenablegroups, syndetic sets, uniformity in recurrence, Furstenberg correspondence principle. and all k ∈ N there exists n ∈ Z such that(1.1) µ ( E ∩ T n ( E ) ∩ · · · ∩ T n ( k − ( E )) > . We call n ∈ Z satisfying (1.1) a multiple recurrence time. We are interested in understandingthe structure of the set of multiple recurrence times in the context of the ergodic Roth theoremfor amenable groups. For countable amenable groups this is the following statement establishedin [7]. Theorem 1.1 (Ergodic Roth theorem for countable amenable groups) . Let Γ be a countableamenable group, ( X , X , µ ) a standard Borel probability space, and T , S : Γ → Aut( X , X , µ ) twomeasure-preserving actions of Γ on ( X , X , µ ) such that T γ ◦ S γ ′ = S γ ′ ◦ T γ for all γ, γ ′ ∈ Γ . Thenthe limit lim n | Φ n | X γ ∈ Φ n µ ( E ∩ T γ ( E ) ∩ S γ T γ ( E )) exists and is positive whenever µ ( E ) > . Recall that a subset of a group is said to be syndetic if finitely many of its translates coverthe whole group. In the case of finitely many commuting actions of a countable abelian group,Furstenberg and Katznelson establish in [23, Theorem 10.1], in particular, that the set of mul-tiple recurrence times is syndetic for any measurable set of positive probability. An analogousproperty is established in [7, Corollary 5.2] for the set of multiple recurrence times in Theorem1.1.In view of Khintchine’s strengthening of Poincaré’s recurrence theorem, Bergelson, Hostand Kra establish in [4] that the sets { n ∈ Z : µ ( E ∩ T n ( E ) ∩ T n ( E )) > µ ( E ) − δ } and { n ∈ Z : µ ( E ∩ T n ( E ) ∩ T n ( E ) ∩ T n ( E )) > µ ( E ) − δ } are syndetic for all δ > Z -actions.An important aspect of the results in [4] is that the lower bound on the probability of the multiplerecurrence event depends only on the measure of E , but is otherwise uniform over all measure-preserving Z -systems. More precisely, for every ε >
0, any measure-preserving Z -system( X , X , µ, T ) and all E ∈ X with µ ( E ) ≥ ε , we have that { n ∈ Z : µ ( E ∩ T n ( E ) ∩ T n ( E )) > ε − δ } and { n ∈ Z : µ ( E ∩ T n ( E ) ∩ T n ( E ) ∩ T n ( E )) > ε − δ } are syndetic for all δ >
0. This lowerbound is also known to be optimal, see [8, Theorem 2.3] by Boshernitzan, Frantzikinakis, andWierdl.Using Host’s magic extensions [25] which are in turn based on the machinery of stated ex-tensions developed by Austin in [1], Chu extends in [10] the Bergelson-Host-Kra results tothe setting of two commuting ergodic measure-preserving actions T , S : Z → Aut( X , X , µ ) Khintchine’s recurrence theorem [32] states that the recurrence set { n ∈ Z : µ ( E ∩ T n ( E )) > µ ( E ) − δ } issyndetic for every δ > NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 3 of the integers on a standard Borel space. Interestingly, [10, Theorem 1.1] establishes that { n ∈ Z : µ ( E ∩ T n ( E ) ∩ S n ( E )) > µ ( E ) − δ } is syndetic for all δ >
0. Notice that the exponenton the latter lower bound is now 4 compared to the exponent of 3 in the single transformationcase. In fact, [10, Theorem 1.2] shows that this is best possible in the setting of two com-muting ergodic Z -actions. The theorem of Chu is then extended by Chu and Zorin-Kranich in[11] to establish an analogous Khintchine-type result for two commuting ergodic actions of acountable amenable group as in Theorem 1.1, using similar techniques of magic and sated ex-tensions. Finally, we also point out the recent paper by Moragues [34], where some versions ofthe Bergelson-Host-Kra uniformity results are obtained for finitely many commuting measure-preserving ergodic Z -actions. We stress that all these uniformity results [4, 10, 11, 34] requirecertain ergodicity hypotheses.In a di ff erent direction, the following uniformity aspect of the Furstenberg-Katznelson mul-tiple recurrence theorem [22] is mentioned in [5, Remark 1.6]. Let R be a set of k -recurrence(see [18] for a list of sets of k -recurrence). Then for any ε > R ′ ⊂ R and δ > X , X , µ, T , . . . , T k ) with finitelymany commuting shifts T , . . . , T k and any E ∈ X with µ ( E ) ≥ ε there exists n ∈ R ′ such that µ ( E ∩ T n ( E ) ∩ . . . ∩ T nk ( E )) > δ .One major objective of this paper is to establish a new uniformity aspect for multiple recur-rence times in the ergodic Roth theorem for amenable groups. To state our result, we need todefine abstract measure-preserving systems and the notion of uniform amenability which is in-troduced by Keller in [31] and which we review in Appendix B. A relevant property of uniformamenability is that it preserves amenability when taking ultralimits [31], see Appendix C fordetails.Let Γ be a discrete group and let ( X , µ ) be a probability algebra. Probability algebras arealso called measure algebras in the ergodic theory literature, e.g., see [21, 24]. They can bethought of as point-free or pointless probability spaces since there is no underlying set a priori .More precisely, a probability algebra is an abstract σ -complete Boolean algebra equipped witha probability measure. For definitions and notations on abstract Boolean algebras we refer toAppendix A and see Section 2 for an introduction to probability algebras and the relation be-tween probability spaces and probability algebras. Since we do not assume that Γ is countable,there is a risk of dealing with uncountably many null sets if we were to work with classicalprobability spaces. We refer the interested reader to [28, 30, 27] for a more detailed discussionabout the point-free approach to the ergodic theory of uncountable group actions. Usually, thereis also no loss of generality working with probability algebras since one can always quotientout the null ideal of a probability space to obtain a probability algebra. We also do not assumethat a probability algebra is separable. Here separability refers to the topology induced by themetric d ( E , F ) : = µ ( E ∆ F ) on X . A separable probability algebra, for example, is obtainedby quotienting out the null ideal from a standard Borel probability space which is the usual P. DURCIK, R. GREENFELD, A. ISELI, A. JAMNESHAN, AND J. MADRID assumption on probability spaces in the ergodic theory literature. We denote by Aut( X , µ ) theautomorphism group of the probability algebra ( X , µ ), see Definition 2.1. A group homomor-phism T : Γ → Aut( X , µ ) is called an abstract action . Given a second abstract action S , we saythat T and S commute if T γ ◦ S γ ′ ( E ) = S γ ′ ◦ T γ ( E ) for all E ∈ X and γ, γ ′ ∈ Γ , where ◦ denotescomposition of functions. The tuple ( X , µ, T , S ) is called an abstract Roth Γ -dynamical system .We establish the following result. Theorem 1.2 (Uniform syndeticity in the ergodic Roth theorem) . Let G = G ( F ) be a uniformlyamenable set of groups and let ε > . Then for every Γ ∈ G , any abstract Roth Γ -dynamicalsystem ( X , µ, T , S ) , and each E ∈ X with µ ( E ) ≥ ε there exist δ, η > , only depending on ε and G , such that BD Γ ( { γ ∈ Γ : µ ( E ∧ T γ ( E ) ∧ S γ T γ ( E )) > δ } ) > η, where BD Γ denotes lower Banach density defined in Definition B.4. As the ergodic Roth theorem for amenable groups in [7] has both countability and separabilityrestrictions, for our uniform syndeticity result to hold in full generality, we also need to establishan uncountable version of the ergodic Roth theorem for amenable groups:
Theorem 1.3 (Ergodic Roth theorem for arbitrary amenable groups) . Let Γ be an arbitraryamenable discrete group. Let ( X , µ, T , S ) be an arbitrary abstract Roth Γ -dynamical system.Then for every E ∈ X and left Følner net ( Φ α ) α ∈ A for Γ , the limit (1.2) lim α ∈ A | Φ α | X γ ∈ Φ α µ ( E ∧ T γ ( E ) ∧ S γ T γ ( E )) exists and is independent of the choice of the left Følner net. Moreover, the limit is positivewhenever µ ( E ) > . The ergodic Roth theorem for countable amenable groups acting on standard Borel probabil-ity spaces is established in [7]. More precisely, [7, Theorem 4.8] proves the existence of thelimit and [7, Theorem 5.2] establishes the positivity statement. The proof of [7, Theorem 4.8] isbased on Furstenberg-Zimmer structure theory [44, 45, 21]. These two results are extended tothe action of locally-compact second countable groups on standard Borel probability spaces byRobertson [37], where he identifies characteristic factors for non-conventional ergodic averagesof, in particular, two commuting actions and proves positivity (see [37, Theorem 7.1]). Zorin-Kranich establishes in [46, Theorem 1.1] a general convergence theorem for non-conventionalergodic averages for finitely many commuting actions of an arbitrary locally compact amenablegroup on an L space by obtaining the necessary modifications in the proof of Walsh’s conver-gence theorem [42]. However the proof of the L convergence result of Zorin-Kranich does notgive any information on the limit. In particular, it does not yield positivity of the limit for non-negative functions with positive mean, and therefore does not establish a corresponding multiple See Appendix B for a definition of amenability and Følner nets.
NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 5 recurrence theorem. In the case of finitely many commuting actions of a countable amenablegroup, this positivity result was established by Austin [2] using his theory of sated extensions.Austin’s paper also gives an alternative proof of L convergence. We are currently working onextending Austin’s work to an uncountable and point-free framework and use this extension toestablish a multiple recurrence theorem and a uniform syndeticity theorem analogous to Theo-rems 1.2 and 1.3 for finitely many commuting actions of arbitrary discrete amenable groups onarbitrary probability algebras.Before discussing the proof of Theorems 1.2 and 1.3, let us locate our uniform syndeticityresult in the context of the aforementioned literature.Note that the results of Bergelson, Host and Kra concern merely the syndeticity of the mul-tiple recurrence sets. By using the notion of lower Banach density to quantify the degree ofsyndeticity, Theorem 1.2 provides a new information on the uniformity aspects of the degreeof syndeticity of the multiple recurrence sets in the case of linear 3-term progressions. To theauthors’ knowledge, our uniformity result is the first of its kind for a single uniformly amenablegroup, countable as well as uncountable, and for a set of uniformly amenable groups. More-over, it is the first result establishing uniform syndeticity for multiple recurrence times; thusestablishing the existence of a lower bound on the degree of syndeticity uniformly over a classof Roth-type measure-preserving dynamical systems for a uniformly amenable set of groups.This result seems to be new already in the case of Furstenberg’s ergodic Roth theorem for Z -actions [20, §3]. Notice that we do not assume ergodicity, an assumption explicitly made inthe aforementioned uniformity literature [4, 10, 11, 34]. The ultralimit analysis’ method of ourproof does not readily give any explicit bounds for δ, η in Theorem 1.2. Under the hypothesisof ergodicity an explicit bound on δ aligning with Khintchine’s recurrence theorem is providedin [4, 10, 11, 34]. However it is shown that such a lower bound fails if one removes ergodicity,e.g., see [4, Theorem 2.1]. We plan to further investigate bounds for δ, η appearing in Theorem1.2 and understand its connections with the counterexamples in [4, 10, 11, 34] when removingthe ergodicity hypothesis in future work.Applying an uncountable Furstenberg correspondence principle, we can also deduce fromTheorem 1.3 that the set of triangular configurations, i.e. ( θ, ζ ) , ( γθ, ζ ) , ( γθ, γζ ), in a densesubset of Γ × Γ is syndetic for any uncountable amenable discrete group Γ .1.1. Proof methods.
Our proof of Theorem 1.2 relies on ultralimit analysis, which is discussedin Appendix C. To prove Theorem 1.2 we proceed by contradiction by assuming that there is asequence ( X n , µ n , T n , S n ) of abstract Roth Γ n -dynamical systems with Γ n ∈ G and E n ∈ X n with µ n ( E n ) ≥ ε such that BD Γ n ( { γ ∈ Γ n : µ ( E n ∧ T γ ( E n ) ∧ S γ T γ ( E n )) > / n } ) ≤ / n . We can then formthe ultralimit system ( X ∗ , µ ∗ , Γ ∗ , T ∗ , S ∗ ) from this sequence of systems (the uniform amenabilityhypothesis is relevant here to verify that the ultralimit group Γ ∗ is amenable). The di ffi cult andcrucial step is to relate the sequence of lower Banach densities ( BD Γ n ) with the lower Banach P. DURCIK, R. GREENFELD, A. ISELI, A. JAMNESHAN, AND J. MADRID density BD Γ ∗ of the ultraproduct group. We establish a useful relation by employing a Loebmeasure construction and applying a Hahn-Banach extension theorem for invariant means dueto Silverman [40, 39]. Then we apply Theorem 1.3 from which the syndeticity of multiple returntimes for the ultralimit system ( X ∗ , µ ∗ , Γ ∗ , T ∗ , S ∗ ) can be deduced, see Corollary 3.6, whichleads to the desired contradiction. Notice that in the ultralimit system ( X ∗ , µ ∗ , Γ ∗ , T ∗ , S ∗ ) theprobability algebra ( X ∗ , µ ∗ ) is almost never separable and the group Γ ∗ is almost never countableeven if ( X n , µ n ) were separable probability algebras and the Γ n were countable groups. ThusTheorem 1.3 is essential to establish uniform syndeticity even for the class of systems where acountable group acts on a separable probability algebra.As for the proof of Theorem 1.3, we adapt the strategy in [7] for the proof of Theorem 1.1.Since the existence of the limit in Theorem 1.3 follows from Zorin-Kranich’s [46, Theorem1.1], we focus on establishing the multiple recurrence statement. Several adaptations of thestrategy in [7] are required in our uncountable, inseparable, and point-free framework, wherein particular a classical disintegration of measures and related tools such as direct integrals ofHilbert bundles are not necessarily available. Instead we can follow the approach recently de-veloped in [27, 28, 29, 30] by the fourth author and Tao. More precisely, we work with thecanonical model of an abstract Roth Γ -system which is a compact Hausdor ff space (in fact, aStonean space) equipped with a Baire-Radon probability measure which is invariant under theaction of Γ by homeomorphsims. We review in Section 2 the construction of the canonicalmodel and mention related references. One immediate useful consequence of the canonicalmodel is that it leads to a canonical disintegration for abstract factor maps. We can use thecanonical disintegration to define relatively independent products. The relatively independentproduct is a relevant construction in order to identify the characteristic factors for the abstractRoth Γ -systems. These characteristic factors are the largest compact factor of the abstract Γ -systems ( X , µ, T ) and ( X , µ, S T ) over the invariant factor of ( X , µ, S ). Since we do not assumethat ( X , µ, S ) is ergodic, these compact factors are not necessarily the Kronecker factors. We callthem the conditional Kronecker factors. In Lemma 3.5 we establish that the conditional Kro-necker factors are characteristic for the abstract Roth-type non-conventional averages. Hencewe can project onto these factors and it su ffi ces to establish the multiple recurrence statementin Theorem 1.3 for these projections. A step in proving the latter multiple recurrence is a finitedimensional approximation of the Γ -orbits of functions in certain finitely generated Γ -invariant L ∞ submodules with respect to the actions of T and S T respectively. In the countable-separableframework of Theorem 1.1 this approximation is achieved by using direct integrals of Hilbertbundles and measurable selection techniques, e.g., see [24, Chapter 9] for a textbook reference.As these tools are not available to us, even after passing to concrete models as these models aretypically highly inseparable and the acting group is still uncountable, we use instead conditionalanalysis techniques as developed in [9, 17, 13]. Our finite dimensional approximation relies ona conditional Gram-Schmidt process and a conditional Heine-Borel covering lemma which we
NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 7 prove in Appendix D whose statements are inspired by earlier results in the conditional analysisliterature.Finally, to deduce the combinatorial consequence about syndeticity of triangular configura-tions in dense subsets of the product Γ × Γ of an arbitrary discrete amenable group, we can adaptthe proof of [7, Theorem 6.2] after we establish a version of the Furstenberg correspondenceprinciple for uncountable amenable discrete groups first.1.2. Organization of the paper.
In Section 2, we introduce the setup for this work. We proveTheorem 1.3 in Section 3 and Theorem 1.2 in Section 4. In Appendix A, we review Booleanalgebras and the Stone representation theorem. In Appendix B, we review amenability, uni-form amenability, and relate amenability to syndeticity. In Appendix C, we record an ultralimitconstruction for a sequence of abstract Roth dynamical systems. Finally in Appendix D, weestablish a conditional Heine-Borel covering lemma needed in the proof of Theorem 1.3.1.3.
Notation and conventions.
Let X be a set. We denote by P ( X ) its power set. If X is finite,we denote by | X | its cardinality. If E ⊂ X is a subset, we denote by 1 E its indicator function.All Følner nets are understood to be left Følner nets. Similarly, syndeticity is understood asleft syndeticity. By symmetry, all related results in this paper remain true if we replace leftFølner nets with right Følner nets and left syndeticity with right syndeticity. Suppose Γ is agroup and X is a set, and T : Γ × X → X is a group action. Then we stipulate the convention T γ ( f ) = f ◦ T γ − for a function f : X → C . This implies that T γ γ ( f ) = f ◦ T γ − γ − = T γ ( T γ ( f ))for all γ , γ ∈ Γ . Due to this property the induced action on some function space is calledantihomomorphic. With this convention we also have T γ (1 E ) = T γ ( E ) for a subset E ⊂ X . Fora compact Hausdor ff space X , we denote by B a ( X ) the Baire σ -algebra of X , i.e. the smallest σ -algebra generated by the real-valued continuous functions.Throughout, all equalities and inequalities between measurable functions and measurable setsare understood in an almost sure sense. As we will deal with di ff erent measure spaces at thesame time (usually factors or extensions of each others), the almost sure statement is understoodwith respect to the obvious probability measure. For example, f = g for two functions on acommon probability space ( X , X , µ ) is understood as µ ( { x : f ( x ) = g ( x ) } ) =
1. Similarly, weunderstand f < g or f ≤ g and E = F or E ⊂ F for E , F ∈ X . If a certain property is said tohold on a measurable set E , then we mean that µ ( E ∆ F ) = F on whichthis property is satisfied, where ∆ denotes symmetric set di ff erence.A cknowledgments RG was partially supported by the Eric and Wendy Schmidt Postdoctoral Award. AI waspartially supported by the Swiss National Science Foundation (project no. 181898). AJ wassupported by DFG-research fellowship JA 2512 / P. DURCIK, R. GREENFELD, A. ISELI, A. JAMNESHAN, AND J. MADRID
The authors would like to express their gratitude to Terence Tao for inspiring this work, forhelpful discussions, and for his encouragement.2. C anonical models and canonical disintegrations
In this section we describe the setup. We introduce canonical models. With their help, wedevelop some basics of abstract measure theory and define canonical disintegrations whichallows us to define relatively independent products of probability algebras. A more extensivetreatment of these topics can be found in [30].Glasner defines in [24, Definition 2.14] a measure-preserving dynamical system to be a tuple( X , µ, Γ ) where ( X , µ ) is a separable probability algebra and Γ is a countable group of automor-phisms of ( X , µ ). In [24, Theorem 2.15.1], Glasner then shows that any such measure-preservingdynamical system can be modeled by a Cantor measure-preserving system ( ˜ X , B o ( ˜ X ) , ˜ µ, ˜ Γ ),where ˜ X = { , } N is the Cantor space, B o ( ˜ X ) is its Borel σ -algebra, ˜ µ is a Borel probabilitymeasure constructed from µ , and ˜ Γ is a countable group of ˜ µ -preserving homeomorphisms of˜ X . To be modeled means here that both systems are isomorphic in the category of probabil-ity algebra dynamical systems (see Definition 2.1 for a definition of this category). See alsoFurstenberg [21, Section 5.2] for a closely related construction.We introduce next the definition of a topological model for measure-preserving dynamicalsystems ( X , µ, Γ ) where ( X , µ ) is a not necessarily separable probability algebra and Γ is a notnecessarily countable group. We call this compact Hausdor ff model canonical since it satisfiessuitable universality properties, we refer the interested reader to [30, Proposition 7.6] for details.Closely related models haven been suggested either implicitly or explicitly at several occasionsin the literature, e.g., see [19, 14] and [30] for a list of other references.We define the three categories of dynamical systems employed in this work. Definition 2.1 (Concrete and abstract measure-preserving dynamical systems) . Let Γ be a dis-crete group.(i) (The category CncPrb Γ of concrete measure-preserving dynamical systems) A concreteprobability space is a triple ( X , X , µ ) where X is a set, X is a σ -algebra of subsetsof X , and µ : X → [0 ,
1] is a countably additive probability measure. A concretemeasure-preserving map from a concrete probability space ( X , X , µ ) to another ( Y , Y , ν )is a measurable function f : X → Y such that f µ = ν where f µ ( E ) : = µ ( f − ( E )) = ν ( E )for all E ∈ Y . We denote by CncPrb the category of concrete probability spaces.For a concrete probability space ( X , X , µ ), we denote by Aut( X , X , µ ) its automor-phism group, i.e. the group of all bi-measurable functions f : X → X such that both f and its inverse f − are measure-preserving. A concrete measure-preserving dynamicalsystem is a tuple ( X , X , µ, T ) where T : Γ → Aut( X , X , µ ) is a group homomorphism NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 9 γ T γ . We call T a concrete action . Given a second concrete measure-preserving dy-namical system ( Y , Y , ν, S ), a concrete measure-preserving function π : X → Y is calleda concrete factor map if S γ ◦ π ( x ) = π ◦ T γ ( x ) for all x ∈ X and every γ ∈ Γ . In this casewe call ( X , X , µ, T ) a concrete extension of ( Y , Y , ν, S ), and ( Y , Y , ν, S ) a concrete factor of ( X , X , µ, S ). We denote by CncPrb Γ the category of concrete measure-preservingdynamical systems.(ii) (The category CHPrb Γ of topological measure-preserving dynamical systems) A com-pact Hausdor ff probability space is a tuple ( X , B a ( X ) , µ ), where X is a compact Haus-dor ff space with Baire σ -algebra B a ( X ) and µ : X → [0 ,
1] is a Baire-Radon probabilitymeasure (a probability measure satisfying the regularity property µ ( E ) = sup { µ ( F ) : F ∈ X , F ⊂ E , F compact G δ } for all E ∈ B a ( X )). A morphism in the category ofcompact Hausdor ff probability spaces is a measure-preserving continuous function. Wename this category CHPrb . Similarly to
CncPrb Γ , we define the dynamical category CHPrb Γ where now the automorphism group Aut( X , B a ( X ) , µ ) consists of measure-preserving homeomorphisms of compact Hausdor ff probability spaces.(iii) (The category PrbAlg op Γ of opposite probability algebra dynamical systems) We define PrbAlg op Γ as the opposite category of the category PrbAlg Γ op of probability algebradynamical systems. So let’s start by defining the category PrbAlg first. A probabilityalgebra is a tuple ( X , µ ) where X is a σ -complete Boolean algebra (see Appendix A fora basic introduction to Boolean algebras) and µ : X → [0 ,
1] is a countably additiveprobability measure, that is, µ ( W ∞ i = E i ) = P ∞ i = µ ( E i ) for every countable family ( E i )of pairwise disjoint elements in X , µ (1) = µ ( E ) = E =
0. Wedefine a probability algebra morphism from a probability algebra ( Y , ν ) to another ( X , µ )to be a Boolean homomorphism f : Y → X such that µ ( f ( E )) = ν ( E ) for all E ∈ Y .Notice that we do not stipulate that f is a Boolean σ -homomorphism since this followsautomatically: If ( E n ) is a countable family of elements of Y with union E = W E n ,then ν ( E \ W Nn = E n ) = µ ( f ( E ) \ W Nn = f ( E n )) → f ( E ) = W f ( E n ) because µ ( E ) = E =
0. Probability algebras and probability algebra morphismsform the category
PrbAlg of probability algebras.The automorphism group of a probability algebra is the group Aut( X , µ ) consistingof all measure-preserving Boolean isomorphisms of X to itself. A probability algebradynamical system is a tuple ( X , µ, T ) where T : Γ op → Aut( X , µ ) is a group homomor-phism γ T γ , where Γ op denotes the opposite group. We call T an abstract action . Amorphism from a probability algebra dynamical system ( Y , ν, S ) to another ( X , µ, T ) is aprobability algebra morphism π : Y → X such that T γ ◦ π ( E ) = π ◦ S γ ( E ) for all E ∈ Y and γ ∈ Γ . We call π also an abstract extension map , Y an abstract factor of X , and The use of the opposite category is motivated by keeping certain functors (e.g., the canonical model functor)covariant, see [30, Figure 8] for an illustration and [30, Definition A.12] for a definition of the opposite category. X an abstract extension of Y . When changing from PrbAlg Γ to the opposite dynamicalcategory PrbAlg op Γ all previous relations automatically reverse.We describe next the two important processes of how to canonically associate an oppositeprobability algebra system to any concrete measure-preserving system, and conversely howto canonically associate to any opposite probability algebra system a topological measure-preserving system. Throughout we fix a discrete group Γ .Let ( X , X , µ, T ) be a CncPrb Γ -system and let N µ = { E ∈ X : µ ( E ) = } denote the idealof null sets of ( X , X , µ ). Then the quotient Boolean algebra X µ : = X / N µ , resulting from iden-tifying E , F ∈ X whenever µ ( E ∆ F ) =
0, is σ -complete and we have a canonical Boolean σ -epimorphism π : X → X µ which associates to each E ∈ X its equivalence class [ E ] in X µ . Wedefine the associated probability algebra measure ¯ µ : X µ → [0 ,
1] by ¯ µ ([ E ]) : = µ ( E ). For any γ ∈ Γ , we define ¯ T γ : X µ → X µ by ¯ T γ ([ E ]) : = π (( T γ ) − ( E )). We obtain a probability algebra ac-tion ¯ T : Γ → Aut( X µ , ¯ µ ) such that ( X µ , ¯ µ, ¯ T ) is a probability algebra dynamical system. Passingto the dual category provides us with a canonical choice of a PrbAlg op Γ -system associated to( X , X , µ, T ). Of course, the same construction works for any CHPrb Γ -system as well. This com-bined abstraction and deletion process, i.e. when we delete the null sets and with it the point-setstructure of the measurable space ( X , X ), is functorial. In particular, any CncPrb Γ -factor map isassociated to a PrbAlg op Γ -factor map (by the use of the opposite category). However this func-tor is not injective on objects; for example the associated probability algebra cannot distinguishbetween a concrete probability space and its measure-theoretic completion.The canonical model functor reverses this process by associating to any PrbAlg op Γ -dynamicalsystem a canonical CHPrb Γ -dynamical system. We sketch one of the two constructions of thecanonical model functor given in [30, Sections 7, 9] which is based on the Stone representationtheorem (the latter theorem is recalled in Appendix A). Let ( X , µ, T ) be a PrbAlg op Γ -dynamicalsystem. Let Conc ( X ) denote the Stone space of the Boolean algebra X and equip it with the Baire σ -algebra B a ( Conc ( X )). We define the measure of E ∈ B a ( Conc ( X )) to be the measure of theunique element of X that generates a clopen subset of Conc ( X ) that di ff ers from E by a Baire-meager set. It can be checked that this meausure is a Baire-Radon probability measure whichwe denote by µ Conc ( X ) . For γ ∈ Γ , we define T γ Conc ( X ) : = Conc ( T γ X ) : Conc ( X ) → Conc ( X ) to bethe unique homeomorphism resulting from applying the Stone functor to the opposite Booleanisomorphism T γ : X → X . In particular, the inverse image of Baire-meager sets under T γ Conc ( X ) are Baire meager and T γ Conc ( X ) preserves the Baire-Radon probability measure µ Conc ( X ) . We ob-tain a CHPrb Γ -system ( Conc ( X ) , B a ( Conc ( X )) , µ Conc ( X ) , T Conc ( X ) ) which is called the canonicalmodel of ( X , µ, T ). This correspondence is again functorial, in particular PrbAlg op Γ -factor mapsare mapped to CHPrb Γ -factor maps. In fact, we have a stronger functorial property: If we com-pose the canonical model functor with the combined abstraction and deletion functor we obtainthe identity functor on PrbAlg op Γ (up to natural isomorphisms). NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 11
We can use the canonical model functor to introduce L p spaces and integration on PrbAlg op -spaces ( X , µ ) by just defining L p ( X ) : = L p ( Conc ( X ))for 1 ≤ p ≤ ∞ , and defining the integral of f ∈ L ( X ) to be R Conc ( X ) f d µ Conc ( X ) . One canalso define abstract L p spaces on probability algebras (or more generally, on measure algebras)directly without invoking a canonical model, see [19]. One can then show that these abstract L p spaces are isomorphic (as Banach and Riesz spaces) to the ones defined above (see [30,Remark 9.13] for a comparison). If ( X , µ, T ) is a PrbAlg op -dynamical system, then we definethe Koopman operator T γ : L p ( X ) → L p ( X ) by T γ ( f ) : = f ◦ T γ − Conc ( X ) , f ∈ L p ( X )for all γ ∈ Γ .Given a CncPrb -space ( X , X , µ ), we also have the identifications L p ( X ) ≡ L p ( X µ ) = L p ( Conc ( X µ ))as Riesz and Banach spaces. We will freely make use of these identifications in the sequel.If π : ( X , µ, T ) → ( Y , ν, S ) is a PrbAlg op Γ -factor map, then we have the pullback map π ∗ : L p ( Y ) → L p ( X ) defined by π ∗ ( f ) : = f ◦ Conc ( π ) which is easily seen to be an isometry. Thuswe can identify L p ( Y ) with the closed invariant subspace π ∗ ( L p ( Y )) in L p ( X ). In the case of p =
2, the pullback map π ∗ induces a conditional expectation operator E ( · | Y ) : L ( X ) → L ( Y )by defining E ( f | Y ) : = E ( f | π ∗ ( L ( Y )))where E ( f | π ∗ ( L ( Y ))) is the orthogonal projection onto π ∗ ( L ( Conc ( Y ))) seen as a closed sub-space of L ( Conc ( X )). Furthermore, we have E ( T γ ( f ) | Y ) = S γ ( E ( f | Y )) for all f ∈ L ( Conc ( X )).As a first example of a PrbAlg op Γ -factor we have the PrbAlg op Γ -invariant factor. Let ( X , µ, T )be a PrbAlg op Γ -dynamical system. Then we define the PrbAlg op Γ -invariant factor to consist ofthe σ -complete Boolean algebra Inv Γ ( X , µ, T ) : = { E ∈ X : T γ ( E ) = E ∀ γ ∈ Γ } equipped with the probability measure µ and the restriction of the action T to Inv Γ ( X , µ, T )which is just the trivial action. We call ( Inv Γ ( X , µ, T ) , µ, T ) the PrbAlg op Γ - invariant factor of( X , µ, T ), where the factor map π : X → Inv Γ ( X , µ, T ) is the canonical projection. The invariantfactor is a functor from the category PrbAlg op Γ to itself. That is, if π : ( X , µ, T ) → ( Y , ν, S ) is a PrbAlg op Γ -extension, then we have an induced PrbAlg op Γ -factor map π : ( Inv Γ ( X , µ, T ) , µ, T ) → ( Inv Γ ( Y , ν, S ) , ν, S ). We can combine the invariant factor functor with the canonical model func-tor to find a canonical representation of the invariant factor and the canonical projection in CHPrb Γ . A PrbAlg op Γ -system ( X , µ, T ) is said to be ergodic if Inv Γ ( X , µ, T ) = { , } is thetrivial algebra. The following result is established in [30, Theorem 1.6], see also [16, §2] and the referencesin [30] for related results in the literature.
Theorem 2.2 (Canonical disintegration) . Let Γ be a discrete group. Let ( X , µ, T ) , ( Y , ν, S ) be PrbAlg op Γ -dynamical systems, and let π : X → Y be a
PrbAlg op Γ -factor map. Then thereis a unique Radon probability measure µ y on Conc ( X ) for each y ∈ Conc ( Y ) which dependscontinuously on y in the vague topology in the sense that y R Conc ( X ) f d µ y is continuous forevery f in the space of continuous functions C ( Conc ( X )) , and such that (2.1) Z Conc ( X ) f ( x ) g ( Conc ( π )( x )) d µ Conc ( X ) ( x ) = Z Conc ( Y ) Z Conc ( X ) f d µ y ! g d µ Conc ( Y ) for all f ∈ C ( Conc ( X )) , g ∈ C ( Conc ( Y )) . Furthermore, for each y ∈ Conc ( Y ) , µ y is supported onthe compact set Conc ( π ) − ( { y } ) , in the sense that µ Y ( E ) = whenever E is a measurable set dis-joint from Conc ( π ) − ( { y } ) . (Note that this conclusion does not require the fibers Conc ( π ) − ( { y } ) to be measurable.) Moreover, we have µ S γ Conc ( Y ) ( y ) = ( T γ Conc ( X ) ) µ y for all y ∈ Conc ( Y ) and γ ∈ Γ . We can use the canonical disintegration to define relatively independent products. Let ( X , µ, T )and ( Y , ν, S ) be PrbAlg op Γ -dynamical systems and π : X → Y a PrbAlg op Γ -factor map. Then wecan define the CHPrb Γ -dynamical system( Conc ( X ) × Conc ( X ) , B a ( Conc ( X ) × Conc ( X )) , µ Conc ( X ) × Conc ( Y ) µ Conc ( X ) , T Conc ( X ) × T Conc ( X ) )as follows. We have the well-known fact that B a ( Conc ( X ) × Conc ( X )) = B a ( Conc ( X )) × B a ( Conc ( X )) , e.g., see [28, Lemma 2.1], and which is the main reason adopting the "Baire-centric" perspec-tive, see [30] for further illustration of this perspective.We define the relatively independent product measure µ Conc ( X ) × Conc ( Y ) µ Conc ( X ) ( E ) = Z Conc ( Y ) µ y × µ y ( E ) d ν Conc ( Y ) for E ∈ B a ( Conc ( X ) × Conc ( X )). Finally, we define the product action T Conc ( X ) × T Conc ( X ) : Γ → Aut(
Conc ( X ) × Conc ( X ) , B a ( Conc ( X ) × Conc ( X )) , µ Conc ( X ) × Conc ( Y ) µ Conc ( X ) ) γ ( T Conc ( X ) × T Conc ( X ) ) γ ( x , y ) : = T γ Conc ( X ) ( x ) × T γ Conc ( X ) ( y ) . This canonical relatively independent product has exactly the same properties as its classicalcounterpart for standard Borel spaces and countable group actions as for example recorded in[21, Section 5.5], see [30, §8]. In particular, we have(2.2) Z Conc ( Y ) E ( f | Y ) E ( g | Y ) d µ Conc ( Y ) = Z Conc ( X ) × Conc ( X ) f × gd µ Conc ( X ) × Conc ( Y ) µ Conc ( X ) . NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 13
3. A n ergodic R oth theorem for uncountable amenable groups , and an application In this section we prove Theorem 1.3 and derive from it a combinatorial application. InSection 3.1, we describe the conditional Kronecker factors which control the convergence ofthe non-conventional ergodic averages as occurring in Theorem 1.3. We verify in Lemma 3.5that these factors are characteristic. We provide the necessary versions of the mean ergodictheorem (Theorem 3.2) and the van der Corput lemma (Lemma 3.3) for uncountable groupactions needed to prove Lemma 3.5. We then prove Theorem 1.3 along the lines of the proofin [7]. Finally, we apply Theorem 1.3 in Section 3.2 to find triangular configurations in densesubsets of arbitrary amenable groups. To this end, we prove a version of the correspondenceprinciple of Furstenberg for arbitrary amenable groups.3.1.
Characteristic factors and non-conventional averages.
The Kronecker factor controlsthe non-conventional ergodic averages in the ergodic Roth theorem [20, §3]. Arguably, thisobservation is the starting point of ergodic Ramsey theory (and of the subsequently developedequivalent structure theories of Host-Kra [26] and Ziegler [43] classifying the characteristicfactors in the ergodic Szemerédi theorem). In an ergodic system the Kronecker factor is thecompact factor of the system and corresponds to the largest group rotation inside the system.In a non-ergodic system the Kronecker factor is the largest compact extension of the invariantfactor.When considering two commuting actions, say S and T , of an amenable group, Bergel-son, McCutcheon and Zhang identified in [7, §4] that the convergence of the correspondingnon-conventional ergodic averages as occurring in Theorem 1.1 is controlled by the compactextensions relative to the S -invariant factor of the two systems corresponding to the T and S T -actions respectively. Then in [7, §2,3] a careful Furstenberg-Zimmer type structural analysis ofextensions of two systems with respect to a common factor is carried out, which is used in [7,§4] to establish the existence of the limit in Theorem 1.1, and in [7, §5] its positivity statement.Since we will use [46, Theorem 1.1] of Zorin-Kranich for the existence part of the limit in ourergodic Roth theorem for uncountable amenable groups (Theorem 1.3), we can largely avoidadapting the relativized analysis in [7, §2,3] and focus on the necessary modifications neededto establish the positivity of the limit in Theorem 1.3.
Remark 3.1.
We remark that the Furstenberg-Zimmer type structural analysis in [7, §2-4] canbe suitably adapted to the setting of an uncountable group acting on an inseparable probabilityalgebra to obtain a novel ergodic theoretic proof of the special case in Zorin-Kranich’s [46,Theorem 1.1] corresponding to the ergodic Roth theorem for discrete amenable groups. Basedon conditional analysis techniques, the fourth author develops in [27] an uncountable version ofthe Furstenberg-Zimmer structure theory based on which such an adaptation could be carriedout. The portion of conditional analysis we need in the proof of the positivity statement ofTheorem 1.3 is developed in Appendix D.
Let us set up the stage. For the remainder of this section fix an arbitrary discrete amenablegroup Γ . A Følner net for Γ is denoted by ( Φ α ) α ∈ A and recall our standing assumption that allFølner nets are understood to be left Følner nets. We start by collecting two well-known results. Theorem 3.2 (Mean ergodic theorem) . Let ( X , µ, T ) be a PrbAlg op Γ -system and f ∈ L ( X ) .Then we have lim α ∈ A | Φ α | X γ ∈ Φ α T γ Conc ( X ) ( f ) = E ( f | Inv Γ ( X , µ, T )) in L ( X ) .Proof. E.g., see [35, Theorem 5.7]. (cid:3)
We give a proof of van der Corput’s lemma for uncountable amenable groups for the sake ofcompleteness by adapting the proof of the countable version in [7].
Lemma 3.3.
Let { f γ : γ ∈ Γ } be a subset of a (not necessarily separable) Hilbert space H suchthat sup γ ∈ Γ k f γ k H < ∞ . If (3.1) lim α ∈ A | Φ α | (cid:18) lim sup β ∈ A | Φ β | X γ ∈ Φ β X η,ρ ∈ Φ α h f ηγ , f ργ i H (cid:19) = , then lim α ∈ A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | Φ α | X γ ∈ Φ α f γ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H = . The lim sup in (3.1) can be defined in a various of equivalent ways. For example, for anet ( x α ) α ∈ A of real numbers, where ( A , ≤ ) is a directed set (that is, a partially ordered set withthe property that for every pair a , b ∈ A there is c ∈ A such that a , b ≤ c ). We can definelim sup α ∈ A x α : = inf α ∈ A sup β ≥ α x β , where { β ≥ α } are the "tails" of the net. Proof.
Let us rewrite1 | Φ β | X γ ∈ Φ β f γ = | Φ β | X γ ∈ Φ β | Φ α | X ω ∈ Φ α f ωγ ! + | Φ β | X γ ∈ Φ β f γ − | Φ β | X γ ∈ Φ β | Φ α | X ω ∈ Φ α f ωγ ! . (3.2)Note that for the second term of the right-hand side of (3.2),(3.3) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | Φ β | X γ ∈ Φ β f γ − | Φ β | X γ ∈ Φ β | Φ α | X ω ∈ Φ α f ωγ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | Φ β | ( X γ ∈ Φ β f γ − | Φ α | X ω ∈ Φ α X γ ∈ ω Φ β f γ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H . Since sup ω ∈ Φ α | Φ β ∆ ω Φ β || Φ β | → β ). Meanwhile, for the first term in the right-hand side of(3.2), the Cauchy-Schwarz inequality yields(3.4) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | Φ β | X γ ∈ Φ β | Φ α | X ω ∈ Φ α f ωγ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H ≤ | Φ β | X γ ∈ Φ β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | Φ α | X ω ∈ Φ α f ωγ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H = | Φ β | X γ ∈ Φ β | Φ α | X ω,ρ ∈ Φ α h f ωγ , f ργ i H . NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 15
Now taking the norm on both sides of (3.2), using the triangle inequality to separate the twoterms on its right-hand side, choosing then first β and second α large enough, the claim followsfrom the hypothesis (3.1). (cid:3) Throughout this subsection, fix a
PrbAlg op Γ -Roth dynamical system ( X , µ, S , T ). Then ( X , µ, S T )is a
PrbAlg op Γ -dynamical system where the abstract action S T : Γ → Aut( X , µ ) is defined by γ S γ ◦ T γ . We write S γ T γ = S γ ◦ T γ . It follows from the commutativity of T , S that( Inv Γ ( X , µ, S ) , µ, T ) is a PrbAlg op Γ -factor of ( X , µ, T ) and ( X , µ, S T ) respectively. In order tolighten the notation, we denote by Y = Inv Γ ( X , µ, S ). Definition 3.4 (Conditional Kronecker factor) . We consider the
PrbAlg op Γ -system ( X , µ, T ).Let H be the closure of the union of all finitely generated, closed, and T -invariant L ∞ ( Y )-submodules of L ( X ). We can identify with H a PrbAlg op Γ -factor ( Z T , µ Z T , T Z T ) of ( X , µ, T ) inthe sense that the pullback of L ( Z T ) in L ( X ) by the factor map is H . We call ( Z T , µ Z T , T Z T ) the conditional Kronecker factor of ( X , µ, T ). Similarly, we define the conditional Kronecker factorof ( Z S T , µ Z ST , T Z ST ) of ( X , µ, S T ) relative to the invariant factor Y .From now on we work with the canonical models of the above systems and the correspond-ing CHPrb Γ -factor relations among them. For example, the canonical model of ( X , µ, S , T ) isdenoted by ( Conc ( X ) , B a ( Conc ( X )) , µ Conc ( X ) , S Conc ( X ) , T Conc ( X ) ) and we have the CHPrb Γ -factormap( Conc ( Z S T ) , B a ( Conc ( Z S T )) , µ Conc ( Z ST ) , ( S T ) Conc ( Z ST ) ) → ( Conc ( Y ) , B a ( Y ) , µ Conc ( Y ) , T Conc ( Y ) ) . Lemma 3.5.
Suppose that f , g ∈ L ∞ ( X ) with f ⊥ L ( Z T ) or g ⊥ L ( Z S T ) , where we view L ( Z T ) and L ( Z S T ) as subspaces of L ( X ) . Then (3.5) lim α ∈ A | Φ α | X γ ∈ Φ α T γ ( f ) S γ T γ ( g ) = in L ( X ) . We adapt the arguments in [7, Theorem 4.3], we include the details for completeness.
Proof.
It is enough to consider the case f ⊥ L ( Z T ). The proof for the case g ⊥ L ( Z S T ) issimilar. Then 1 | Φ α | X γ ∈ Φ α h u ζγ , u θγ i L ( X ) = | Φ α | X γ ∈ Φ α Z T ζγ ( f ) S ζγ T ζγ ( g ) T θγ ( f ) S θγ T θγ ( g )d µ Conc ( X ) = | Φ α | X γ ∈ Φ α Z T ζ ( f ) T θ ( f ) S ζγ T ζ ( g ) S θγ T θ ( g )d µ Conc ( X ) = Z T ζ ( f ) T θ ( f ) | Φ α | X γ ∈ Φ α S ζ T ζ S γ ( g ) S θ T θ S γ ( g ) d µ Conc ( X ) . The second equality is due to T γ being measure preserving. By Theorem 3.2 and orthogonaldecomposition, we have1 | Φ α | X γ ∈ Φ α h u ζγ , u θγ i L ( X ) → Z T ζ ( f ) T θ ( f ) E ( S ζ T ζ ( g ) S θ T θ ( g ) | Y )d µ Conc ( X ) = Z E ( T ζ ( f ) T θ ( f ) | Y ) E ( S ζ T ζ ( g ) S θ T θ ( g ) | Y )d µ Conc ( Y ) . By (2.2) this equals Z T ζ × S ζ T ζ ( f × g ) T θ × S θ T θ ( f × g ) d µ Conc ( X ) × Conc ( Y ) µ Conc ( X ) = : a ζ,θ . Note that by Lemma 3.3, to conclude the proof it su ffi ces to show thatlim α ∈ A | Φ α | X ζ,θ ∈ Φ α a ζ,θ = . Observe that the left hand-side of the last display equalslim α ∈ A k | Φ α | X γ ∈ Φ α T γ Conc ( X ) × S γ Conc ( X ) T γ Conc ( X ) ( f × g ) k L ( X ) . This equals zero by another application of Theorem 3.2 since f × g is orthogonal to the in-variant factor of the relatively independent product with respect to the action of T Conc ( X ) × S Conc ( X ) T Conc ( X ) . The proof is complete. (cid:3) We now prove our ergodic Roth theorem for uncountable amenable groups.
Proof of Theorem 1.3.
Recall that the existence of the limit (1.2) follows from [46, Theorem1.1]. Therefore, it remains to establish positivity of (1.2). More precisely, for every E ∈B a ( Conc ( X )) with µ Conc ( X ) ( E ) > α ∈ A | Φ α | X γ ∈ Φ α µ Conc ( X ) ( E ∩ T γ Conc ( X ) ( E ) ∩ S γ Conc ( X ) T γ Conc ( X ) ( E )) > . By Lemma 3.5 and an orthogonal decomposition, it is enough to show thatlim α ∈ A | Φ α | X γ ∈ Φ α Z Conc ( X ) E E (1 E | Z T ) E (1 E | Z S T ) d µ Conc ( X ) > . Since µ Conc ( X ) ( E ) >
0, it holds that E (1 E | Z T ) E (1 E | Z S T ) > E . Thus, there exist r > E ′ ∈ B a ( Conc ( X )) with E ′ ⊂ E and µ Conc ( X ) ( E ′ ) > E (1 E | Z T ) E (1 E | Z S T ) > r on E ′ . Recall that we denote by Y the invariant factor of ( X , µ, S ). Since0 < µ Conc ( X ) ( E ′ ) = Z Conc ( Y ) E (1 E ′ | Y ) d µ Conc ( Y ) , NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 17 we find t > F ∈ B a ( Conc ( Y )) with µ Conc ( Y ) ( F ) > E (1 E ′ | Y ) > t on F . Inparticular, we obtain E (1 E E (1 E | Z T ) E (1 E | Z S T ) | Y ) ≥ r E (1 E ′ | Y ) > rt on F . Furthermore, by the definition of a compact extension, there exist sequences ( f n ) and ( g n ) suchthat each f n is contained in a T -invariant finitely generated L ∞ ( Y )-submodule of L ( X ), each g n is contained in an ( S T )-invariant finitely generated L ∞ ( Y )-submodule of L ( X ), and k E (1 E | Z T ) − f n k L ( X ) → k E (1 E | Z S T ) − g n k L ( X ) → . We can and will assume that f n and g n are bounded for all n . Indeed, we can define f n , m = f n E ( | f n | | Y ) ≤ m . Then f n , m is an element of the same L ∞ ( Y )-submodule as f n for each m and every n and the diagonal sequence ( f n , n ) approximates E (1 E | Z T ) in L ( X ). Similarly for ( g n ).Using the identity k u k L ( X ) = Z Conc ( Y ) E ( | u | | Y ) d µ Conc ( Y ) and passing to a subsequence if necessary, we have that E ( | E (1 E | Z T ) − f n | | Y ) → E ( | E (1 E | Z S T ) − g n | | Y ) → F ′ ∈ B a ( Conc ( Y )) with F ′ ⊂ F and µ Conc ( Y ) ( F ′ ) > µ Conc ( Y ) ( F ′ ) > µ Conc ( Y ) ( F ) and E ( | E (1 E | Z T ) − f n | | Y ) → E ( | E (1 E | Z S T ) − g n | | Y ) → F ′ . In particular, for any ε > f and g in a T - and ( S T )-invariant finitely generated L ∞ ( Y )-submodule of L ( X ), respectively, such that(3.7) E ( | E (1 E | Z T ) − f | | Y ) < ε and E ( | E (1 E | Z S T ) − g | | Y ) < ε on F ′ . Since both f , g lie in finitely generated, closed, and T - and ( S T )-invariant L ∞ ( Y )-submodulesof L ( X ), respectively, using Lemma D.1 we find h , . . . , h l ∈ L ( X ) such that for each γ ∈ Γ ,(3.8) E ( | T γ Conc ( X ) ( f ) − h N T γ | | Y ) < ε and E ( | S γ Conc ( X ) T γ Conc ( X ) ( g ) − h N ST γ | | Y ) < ε on F ′ , where N T γ = l X m = m C m ,γ , N S T γ = l X m = m D m ,γ , and ( C m ,γ ) and ( D m ,γ ) are defined as follows. Let ˜ C m ,γ = { E ( | T γ Conc ( X ) ( f ) − h m | | Y ) < ε } for m = , . . . , l , and set C ,γ = ˜ C ,γ and C m ,γ = ˜ C m ,γ \ S m − m ′ = ˜ C m ′ ,γ for m = , . . . , l . Similarly define D m ,γ , m = , . . . , l with T γ Conc ( X ) ( f ) replaced by S γ Conc ( X ) T γ Conc ( X ) ( g ).Next we show the following claim. Claim.
Let M >
0. For any 0 < δ < γ , . . . , ˜ γ M ∈ Γ such that(i) µ Conc ( X ) ( F ′ ∩ ( T ˜ γ Conc ( X ) ) − F ′ ∩ . . . ∩ ( T ˜ γ M Conc ( X ) ) − F ′ ) > ( δµ Conc ( X ) ( F ′ )) M ;(ii) ˜ γ − i ˜ γ j ∈ Φ α for some α ∈ A whenever 1 ≤ i < j ≤ M . We identify a subset Y ′ of the factor Y with the subset π − ( Y ′ ) of X , where π : ( X , µ, S ) → Y is the factor map. Proof of claim.
Observe that by Theorem 3.2,lim α ∈ A | Φ α | X γ ∈ Φ α µ Conc ( X ) ( F ′ ∩ ( T γ Conc ( X ) ) − ( F ′ )) = h F ′ , E [1 F ′ | Z T ] i L ( X ) (3.9)This equals k E [1 F ′ | Z T ]] k L ( x ) , which is bounded from below by k E [1 F ′ | Z T ]] k L ( x ) = µ Conc ( X ) ( F ′ ) . By the pigeonhole principle, for every 0 < δ < α ∈ A and γ ∈ Φ α such that µ Conc ( X ) ( F ′ ∩ ( T γ Conc ( X ) ) − ( F ′ )) > δ ( µ Conc ( X ) ( F ′ )) . For a given Følner net ( Φ α ) α ∈ A and any finite family γ , . . . , γ k ∈ Γ , ( Φ α ∩ γ Φ α ∩ . . . ∩ γ k Φ α ) α ∈ A is also a Følner net. Thus, we may iterate the previous argument.Repeating the previous argument, where we replace F ′ by F = F ′ ∩ ( T γ Conc ( X ) ) − ( F ′ ), we find α ∈ A and γ ∈ Φ α ∩ γ − Φ α such that µ Conc ( X ) ( F ∩ ( T γ Conc ( X ) ) − ( F )) > δ ( µ Conc ( X ) ( F )) . This implies µ Conc ( X ) ( F ′ ∩ ( T γ Conc ( X ) ) − ( F ′ ) ∩ ( T γ Conc ( X ) ) − ( F ′ ) ∩ ( T γ γ Conc ( X ) ) − ( F ′ )) > δ µ Conc ( X ) ( F ′ ) . Set F = F ∩ ( T γ Conc ( X ) ) − ( F ). Choose α ∈ A and γ ∈ Φ α ∩ γ − Φ α ∩ γ − Φ α ∩ ( γ γ ) − Φ α such that µ Conc ( X ) ( F ∩ ( T γ Conc ( X ) ( F )) > δµ Conc ( X ) ( F ) . After M iterations we obtain ˜ γ , . . . , ˜ γ M with the properties (i) and (ii), where ˜ γ i = γ γ · · · γ i for 1 ≤ i ≤ M . This completes the proof of the claim. (cid:3) Now we choose M = l +
1. By the above claim we find ˜ γ , . . . , ˜ γ M with the properties (i)and (ii). By the pigeonhole principle, one can construct measurable functions I , J : Conc ( Y ) →{ , . . . , M } such that I ( y ) < J ( y ), N T ˜ γ I ( y ) ( y ) = N T ˜ γ J ( y ) ( y ), and N S T ˜ γ I ( y ) ( y ) = N S T ˜ γ J ( y ) ( y ). Using (3.8) and thetriangle inequality, we obtain E ( | T ˜ γ I Conc ( X ) ( f ) − T ˜ γ J Conc ( X ) ( f ) | | Y ) < ε on F ′ and E ( | S ˜ γ I Conc ( X ) T ˜ γ I Conc ( X ) ( g ) − S ˜ γ J Conc ( X ) T ˜ γ J Conc ( X ) ( g ) | | Y ) < ε on F ′ . These inequalities imply E ( | f − T ˜ γ − I ˜ γ J Conc ( X ) ( f ) | | Y ) ◦ T ˜ γ − I < ε on F ′ . (3.10) E ( | g − S ˜ γ − I ˜ γ J Conc ( X ) T ˜ γ − I ˜ γ J Conc ( X ) ( g ) | | Y ) ◦ T ˜ γ − I Conc ( X ) < ε on F ′ . In the second inequality we have used that S acts trivially on Y . Let us set D : = F ′ ∩ T ˜ γ Conc ( X ) F ′ ∩ . . . ∩ T ˜ γ M Conc ( X ) F ′ . NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 19
Now for y ∈ D , we have ( T ˜ γ I ( y ) Conc ( X ) ) − ( y ) ∈ F ′ . Therefore, by (3.7) we obtain T ˜ γ I ( y ) Conc ( X ) ( E ( | E (1 E | Z T ) − f | | Y ))( y ) < ε (3.11) T ˜ γ I ( y ) Conc ( X ) ( E ( | E (1 E | Z S T ) − g | | Y ))( y ) < ε. for almost every y ∈ D . Moreover, since also ( T ˜ γ J ( y ) Conc ( X ) ) − ( y ) ∈ F ′ , it holds T ˜ γ I Conc ( X ) ( E ( | T ˜ γ − I ˜ γ J Conc ( X ) ( E (1 E | Z T ) − T ˜ γ − I ˜ γ J Conc ( X ) ( f ) | | Y ) = T ˜ γ J Conc ( X ) E ( | E (1 E | Z T ) − f | | Y ) < ε (3.12)on D . Applying the triangle inequality to (3.10), (3.11), and (3.12), we obtain T ˜ γ I Conc ( X ) E ( | E (1 E | Z T ) − T ˜ γ − I ˜ γ J Conc ( X ) ( E (1 E | Z T )) | | Y ) < ε on D . Similarly, we have T ˜ γ I Conc ( X ) E ( | E (1 E | Z S T ) − T ˜ γ − I ˜ γ J Conc ( X ) ( E (1 E | Z S T )) | | Y ) < ε on D . Let ˜ γ : = ˜ γ ( y ) = ˜ γ − I ( y ) ˜ γ J ( y ) . Setting ε = rt , we then obtain E (1 E T ˜ γ Conc ( X ) ( E (1 E | Z T )) S ˜ γ Conc ( X ) T ˜ γ Conc ( X ) ( E (1 E | Z S T )) | Y ) > E (1 E E (1 E | Z T ) E (1 E | Z S T ) | Y ) − ε > rt D . Let η = ( δµ Conc ( X ) ( F ′ )) M . We can choose B ⊂ D with µ Conc ( X ) ( B ) > η M such that ˜ γ = ˜ γ ( y )is constant on B . Then Z Conc ( X ) E T ˜ γ Conc ( X ) ( E (1 E | Z T )) S ˜ γ Conc ( X ) T ˜ γ Conc ( X ) ( E (1 E | Z S T )) > rt η M > . We have that ˜ γ ∈ Φ α for some α ∈ A . It follows from Proposition B.3 that G = (cid:26) ˜ γ ∈ Γ : Z Conc ( X ) E T ˜ γ Conc ( X ) ( E (1 E | Z T )) S ˜ γ Conc ( X ) T ˜ γ Conc ( X ) ( E (1 E | Z S T )) > rt η M (cid:27) is syndetic, since neither of the quantities E (1 E | Z T ) , E (1 E | Z S T ) , r , t , η, M depend on the choice ofthe Følner net. It follows from Lemma B.5 that BD Γ ( G ) >
0. Thuslim α ∈ A | Φ α | X γ ∈ Φ α Z Conc ( X ) E T ˜ γ Conc ( X ) ( E (1 E | Z T )) S ˜ γ Conc ( X ) T ˜ γ Conc ( X ) ( E (1 E | Z S T )) > ¯ d Φ ( G ) rt η M > , where ¯ d Φ ( G ) = lim sup α | G ∩ Φ α || Φ α | . We therefore obtain (3.6), as needed. (cid:3) We have the following important corollary about the largeness of the set of return times.
Corollary 3.6.
Suppose that E ∈ X with µ ( E ) > . Then there exists δ > such that { γ ∈ Γ : µ ( E ∧ T γ E ∧ S γ T γ E ) > δ } is syndetic.Proof. Assume towards a contradiction that for each n ∈ N , G n = { γ ∈ Γ : µ ( E ∧ T γ E ∧ S γ T γ E ) > / n } is not syndetic. Let ( Φ α ) α ∈ A be a Følner net. Since G n is not syndetic there exists γ n α ∈ Γ \ Φ − α G n for each α ∈ A , n ∈ N . We have that ( Φ n α ) α ∈ A defined by Φ n α : = Φ α γ n α is another Følner net foreach n ∈ N such that Φ n α ∩ G n = ∅ for all α ∈ A , n ∈ N . By construction, for every α ∈ A ,(3.13) 1 | Φ n α | X γ ∈ Φ n α µ ( E ∧ T γ E ∧ S γ T γ E ) ≤ n . By [46, Theorem 1.1 (2)], the limitlim α ∈ A | Φ n α | X γ ∈ Φ n α µ ( E ∧ T γ E ∧ S γ T γ E )exists and is independent of the choice of the Følner net. Moreover, by (3.13), this limit is lessthan any ε >
0, hence it must be zero. However, this contradicts Theorem 1.3. (cid:3)
Triangular patterns in Γ × Γ . In this section we will employ Theorem 1.3 and Corol-lary 3.6 to derive a combinatorial application analogous to [7, Theorem 6.1].By [35, Proposition 0.16 (5)], Γ × Γ is a discrete amenable group if Γ is so as well. We wantto establish the following combinatorial application. Theorem 3.7.
Let Γ be a discrete amenable group and m : ℓ ∞ ( Γ × Γ ) → R be an invariantmean. Suppose that E ⊂ Γ × Γ satisfies m (1 E ) > . Then the set { γ ∈ Γ : there exists ( θ, ζ ) ∈ Γ × Γ such that ( θ, ζ ) , ( γθ, ζ ) , ( γθ, γζ ) ∈ E } is syndetic. To prove Theorem 3.7, we proceed similarly to [7]. We will need a correspondence principleanalogous to [7, Proposition 6.2]. In [6, Theorem 2.1], Bergelson and McCutcheon establish acorrespondence principle for countable amenable semigroups. We adapt this proof for uncount-able discrete amenable groups.Let
Ω = { , } Γ × Γ . Then Ω is a totally disconnected compact Hausdor ff space, thus a Stonespace (see Appendix A). An element ω ∈ Ω corresponds uniquely to a subset of Γ × Γ . We definean action of Γ × Γ on Ω as follows. First let S : Γ → Aut( Ω ) be defined by S γ ( ω )( θ, ζ ) : = ( θγ, ζ )and T : Γ → Aut( Ω ) be defined by T γ ( ω )( θ, ζ ) : = ( θ, ζγ ). Since T , S are commuting, so we candefine U : Γ × Γ → Aut( Ω ) , U : = S ◦ T . Lemma 3.8 (An uncountable Furstenberg correspondence principle) . Fix an invariant meanm : ℓ ∞ ( Γ × Γ ) → R and let E ⊂ Γ × Γ be such that m (1 E ) > . Let X be the U-orbit closureof E in Ω , i.e. X : = { S θ T γ (1 E ) : θ, γ ∈ Γ } . Then there exits a CHPrb Γ -Roth dynamical system ( X , B a ( X ) , µ, S , T ) such that µ ( { ω ∈ X : ω ( e , e ) = } ) > where e is the identity element of thegroup Γ and ω ( e , e ) is the evaluation of ω ∈ Ω at the entry ( e , e ) . NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 21
Proof.
The collection O of cylinder sets(3.14) { x ∈ X : x ( γ ) = a , . . . , x ( γ k ) = a k } (where k ∈ N , γ i ∈ Γ × Γ , a i ∈ { , } , 1 ≤ i ≤ k ) is a clopen base of the topology of X .Let A be the Boolean algebra generated by O and X be the corresponding σ -algebra. By theStone-Weierstraß theorem, we have X = B a ( X ). For a cylinder set D of the form (3.14), define µ ( D ) : = m (1 γ − E · . . . · γ − k E k ) , where E i = E if a i = E i = E c if a i =
0, 1 ≤ i ≤ k . By compactness, µ is a premeasure on A and thus can be extended to a Baire probability measure on X by the Carathéodory extensiontheorem. Any Baire probability measure on a compact Hausdor ff space is Radon (e.g., see [30,Proposition 4.2(iii)]). By construction, µ is T - and S -invariant and satisfies µ ( { ω ∈ X : ω ( e , e ) = } ) = m (1 E ) > . (cid:3) Proof of Proposition 3.7.
By Lemma 3.8, there exists a U -invariant Baire probability measure µ on X : = { S θ T γ E : θ, γ ∈ Γ } such that(3.15) µ ( U γ ( A ) ∩ . . . ∩ U γ n ( A )) = m (1 γ − E · . . . · γ − n E ) , where A = { ω ∈ X : ω ( e , e ) = } . We now pass from the CHPrb Γ -Roth dynamical sys-tem ( X , X , µ, S , T ), for which (3.15) holds, to the corresponding PrbAlg op Γ -dynamical system( X µ , ¯ µ, ¯ S , ¯ T ) by the deletion and abstraction process described in Section 2. This allows us toapply Corollary 3.6 and thereby obtain that Θ : = { γ ∈ Γ : ¯ µ ([ A ] ∧ ¯ T γ ([ A ]) ∧ ¯ S γ ¯ T γ ([ A ])) > } is syndetic in Γ . Thus for every γ ∈ Θ we can choose ξ ∈ A ∩ T γ ( A ) ∩ S γ T γ ( A ) . Since A is open and ξ ∈ { U ( θ,ζ ) (1 E ) : ( θ, ζ ) ∈ Γ × Γ } , there exists ( θ, ζ ) ∈ Γ × Γ such that S θ T ζ (1 E ) = U ( θ,ζ ) (1 E ) ∈ A ∩ T γ ( A ) ∩ S γ T γ ( A ) . Therefore ( θ, ζ ) , ( γθ, ζ ) , ( γθ, γζ ) ∈ E . This finishes the proof. (cid:3)
4. U niform syndeticity in the ergodic R oth theorem for amenable groups In this section, we prove Theorem 1.2 whose proof applies tools from ultralimit analysis, seeAppendix C. The following two lemmas establish the necessary relations between a sequenceof lower Banach densities of possibly di ff erent groups and the lower Banach density of theirultraproduct group. Lemma 4.1.
Let G be a uniformly amenable set of groups. Let ( Γ n ) be a sequence in G . Let pbe a non-principal ultrafilter on N and Γ ∗ = Q n → p Γ n the ultraproduct of ( Γ n ) . Let ( m n ) be asequence of invariant finitely additive probability measures m n : P ( Γ n ) → [0 , . Then we canassociate to ( m n ) an invariant finitely additive probability measures m : P ( Γ ∗ ) → [0 , .Proof. By Łos’s theorem, the Loeb measure m ( A ∗ ) ≔ st (lim n → p m n ( A n ))is an invariant finitely additive probability measure on the algebra of internal subsets A ∗ = Q n → p A n of Γ ∗ where A n ⊂ Γ n for each n . Define M (1 A ∗ ) ≔ m ( A ∗ ), and extend M to the closedlinear hull D of { A ∗ : A ∗ ⊂ Γ ∗ internal } in ℓ ∞ ( Γ ∗ ) by linearity and continuity. The closedsubspace D majorizes ℓ ∞ ( Γ ∗ ) in the sense that for every f ∈ ℓ ∞ ( Γ ∗ ) there exists g ∈ D such that f ≤ g (we can take g = k f k ∞ since 1 ∈ D ). By Silverman’s Hahn-Banach extension theorem forinvariant means [39, 40], we can extend M to an invariant mean on the whole space ℓ ∞ ( Γ ∗ ). (cid:3) Lemma 4.2.
Let G be a uniformly amenable set of groups. Let ( Γ n ) be a sequence in G . Let pbe a non-principal ultrafilter on N and Γ ∗ = Q n → p Γ n the ultraproduct of ( Γ n ) . Let A = Q n → p A n be an internal subset of Γ ∗ , where A n ⊂ Γ n for each n. Then we have (4.1) st (lim n → p BD Γ n ( A n )) = inf { st (lim n → p m n ( A n )) : m n : P ( Γ n ) → [0 , , n ∈ N } , where the m n are invariant finitely additive probability measures.Proof. We show the “ ≤ ”-direction for the equality (4.1). By definition, BD Γ n ( A n ) ≤ m n ( A n ) for allinvariant finitely additive probability measures m n on P ( Γ n ) for all n . Hence, lim n → p BD Γ n ( A n ) ≤ lim n → p m n ( A n ) for any sequence ( m n ) with m n : P ( Γ n ) → [0 ,
1] an invariant finitely additiveprobability measure, and thus st (lim n → p BD Γ n ( A n )) ≤ st (lim n → p m n ( A n )). Taking the infimumon the right-hand side proves the “ ≤ ”-direction.Conversely, for the sake of contradiction, suppose that we had C < D where C , D denote theleft-hand side and right-hand side of the equality (4.1) respectively. Then we have { n ∈ N : BD Γ n ( A n ) < D } ∈ p . This implies that { n ∈ N : ∃ m n : P ( Γ n ) → [0 ,
1] such that m n ( A n ) < D } ∈ p . But this would imply { n ∈ N : m n ( A n ) < st (lim n → p m n ( A n )) } ∈ p which, using the intersection property of ultrafilters, would yield { n ∈ N : m n ( A n ) < m n ( A n ) } ∈ p which is impossible since ∅ < p . This proves “ ≥ ” in (4.1). (cid:3) This extension is not unique in general, however this will not cause an issue later.
NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 23
We are now in a good position to prove Theorem 1.2.
Proof.
Let G be a uniformly amenable set of groups. Suppose by contradiction that thereexists ε > n there are Γ n ∈ G , a PrbAlg op Γ n -Roth dynamical system( X n , µ n , T X n , S X n ) with canonical concrete CHPrb Γ n -representation( Conc ( X n ) , B a ( Conc ( X n )) , µ Conc ( X n ) , T Conc ( X n ) , S Conc ( X n ) ) , and E n ∈ B a ( Conc ( X n )) with µ Conc ( X n ) ( E n ) ≥ ε such that(4.2) BD Γ n ( { γ ∈ Γ n : µ Conc ( X n ) ( E n ∩ T γ Conc ( X n ) E n ∩ S γ Conc ( X n ) T γ Conc ( X n ) E n ) > / n } ) ≤ / n . Fix a non-principal ultrafilter p on N . Construct the ultraproduct PrbAlg op Γ ∗ -Roth dynamicalsystem ( X µ , µ X µ , T , S ) from the sequence ( Conc ( X n ) , B a ( Conc ( X n )) , µ Conc ( X n ) , T Conc ( X n ) , S Conc ( X n ) )by the recipe in Appendix C. By construction, we have µ X µ ([ E ∗ ]) ≥ ε where E ∗ = Q n → p E n . ByCorollary 3.6, there exists δ > D ≔ BD Γ ∗ ( { γ ∗ ∈ Γ ∗ : µ X µ ([ E ∗ ] ∧ T γ ∗ ∗ [ E ∗ ] ∧ S γ ∗ ∗ T γ ∗ ∗ [ E ∗ ]) > δ } ) > . Unwrapping all definitions, we have { γ ∗ ∈ Γ ∗ : µ X µ ([ E ∗ ] ∧ T γ ∗ ∗ [ E ∗ ] ∧ S γ ∗ ∗ T γ ∗ ∗ [ E ∗ ]) } (4.4) = Y n → p { γ n ∈ Γ n : µ Conc ( X n ) ( E n ∩ T γ n Conc ( X n ) E n ∩ S γ n Conc ( X n ) T γ n Conc ( X n ) E n ) } . Denote by A n = { γ n ∈ Γ n : µ Conc ( X n ) ( E n ∩ T γ n Conc ( X n ) E n ∩ S γ n Conc ( X n ) T γ n Conc ( X n ) E n ) } . It follows fromLemma 4.1 and (4.4) that D ≤ inf { st (lim n → p m n ( A n )) : m n : P ( Γ n ) → [0 , , n ∈ N } , (where the m n denote finitely additive invariant probability measures). By Lemma 4.2,(4.5) D ≤ st (lim n → p BD Γ n ( { γ n ∈ Γ n : µ Conc ( X n ) ( E n ∩ T γ n Conc ( X n ) E n ∩ S γ n Conc ( X n ) T γ n Conc ( X n ) E n ) > δ } )) . By hypothesis (4.2), the set R = ( n ∈ N : BD Γ n ( { γ n ∈ Γ n : µ Conc ( X n ) ( E n ∩ T γ n Conc ( X n ) E n ∩ S γ n Conc ( X n ) T γ n Conc ( X n ) E n ) > δ } ) ≤ n ) contains all but finitely many n . Since the Fréchet filter is contained in any non-principal ultra-filter (see the beginning of Appendix C), we have R ∈ p . Therefore it follows from (4.5) that D must be zero, however this contradicts (4.3). (cid:3) A ppendix A. B oolean algebras and the S tone representation theorem A Boolean algebra is a ring ( X , + , · ) with a multiplicative identity 1 in which x = x for every x ∈ X . We always assume the non-degeneracy assumption 0 ,
1. A prototypical example is( X , ∆ , ∩ ) where X is any set and X ⊂ X is an algebra of subsets of X , and ∆ is the symmetricdi ff erence. Its zero is the empty set ∅ and the multiplicative identity is X . In particular, we havethe trivial algebra ( {∅ , X } , ∆ , ∩ ) which is ring-isomorphic to the finite field ( F , + , · ). Given a boolean algebra ( X , + , · ) and x , y ∈ X , we set x ∆ y = x + y , x ∧ y = x · y , x ∨ y = x + y + xy anddenote x ≤ y if and only if x · y = x .A subalgebra is a subring of X which contains its multiplicative identity. A set I ⊂ X is anideal if and only if 0 ∈ I , x ∨ y ∈ I for all x , y ∈ I , and x ∈ I whenever x ≤ y and y ∈ I . Note thatwhile an ideal in a Boolean algebra is necessarily a subring, it constitutes a subalgebra only ifit is X itself. Moreover, the quotient ring X / I is a Boolean algebra called the quotient algebra .A map f : X → Y between two Boolean algebras X and Y is called a Boolean homomorphism if it is a ring homomorphism, i.e. f ( x ∆ y ) = f ( x ) ∆ f ( y ) and f ( x ∧ y ) = f ( x ) ∧ f ( y ), and maps themultiplicative identity of X to the multiplicative identity of Y . Note that f ( X ) is a subalgebra of Y . A Boolean algebra is called σ -complete if every non-empty countable subset has a least upperbound. An ideal I of a Boolean algebra is called a σ -ideal if every non-empty countable subsetof I has a least upper bound in I . If I is a σ -ideal I in a σ -complete Boolean algebra X , then thequotient algebra X / I is σ -complete as well.Any abstract Boolean algebra can be represented by a concrete Boolean algebra of sets byStone’s representation theorem, as follows. Consider the set Z X of all (non-zero) ring homomor-phisms from X to F . The image s ( X ) under the map s : X → X , x s ( x ) = { f ∈ Z : f ( x ) = } is a base of a topology on Z X . The set Z X equipped with the topology generated by s ( X ) is calledthe Stone space of the Boolean algebra X . Moreover one can show that s ( X ) corresponds to theset of all clopen (closed and open) subsets of Z X . Therefore Z X is a totally disconnected space.Moreover one can show that it is also compact Hausdor ff . By Stone’s representation theorem ,the set of all clopen subsets of Z X equipped with the usual set operations is a Boolean alge-bra isomorphic to X , see [33, Section 7] for a comprehensive introduction into this topologicalversion of Stone duality. The Stone space Z X can also be regarded as a closed subspace of ageneralized Cantor space. More precisely, Z X is a closed subspace of { , } X = F X viewed astopological product space with { , } endowed with the discrete topology (see [33, Section 7]for details). Any Boolean homomorphism f : X → Y between Boolean algebras X and Y canbe uniquely represented as a continuous function ˆ f : Z Y → Z X given by ˆ f ( α ) = α ◦ f where α ∈ Z Y is a Boolean homomorphism from Y to F . This correspondence is a contravariant func-tor between the category Boolean algebras and Stone spaces, which establishes a well-knownequivalence of categories, known as Stone duality.A ppendix B. A menability and syndeticity
B.1.
Amenability.
A discrete group Γ is said to be amenable if it satisfies one of the followingequivalent conditions: NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 25 (i) (
Følner condition ) For every finite set Ψ ⊂ Γ and every 0 < ε < Φ = Φ ( ε, Ψ ) ⊂ Γ such that max γ ∈ Ψ | Φ∆ γ Φ | ≤ ε | Φ | . (ii) There exists a Følner net for Γ , that is, a net ( Φ α ) α ∈ A of non-empty finite subsets of Γ ,such that lim α ∈ A | Φ α ∆ γ Φ α || Φ α | → γ ∈ Γ .(iii) There exists an invariant mean for Γ , that is a positive linear functional m : ℓ ∞ ( Γ ) → R with the properties that L (1) = m ( γ f ) = f , where ( γ f )( γ ′ ) : = f ( γ − γ ′ ) is theleft-regular representation.(iv) There exists an invariant finitely additive probability measure µ : P ( Γ ) → [0 , µ : P ( Γ ) → [0 ,
1] such that µ ( γ E ) = µ ( E ) for all γ ∈ Γ and E ⊂ Γ , where γ E = { γ ˜ γ : ˜ γ ∈ E } .See [36] for a proof of these equivalences.B.2. Uniform amenability.
The notion of uniform amenability was introduced by Keller in[31]. This notion is uniform version of the Følner condition, as follows.
Definition B.1 (Uniform amenability) . A discrete group Γ is said to be uniformly amenable ifthere exists a function F : N × (0 , → N such that for every set Ψ ⊂ Γ with | Ψ | ≤ n and0 < ε < Φ ⊂ Γ with | Φ | ≤ F ( n , ε ) such thatmax γ ∈ Ψ | Φ∆ γ Φ | ≤ ε | Φ | . More generally, a set G of discrete groups is said to be uniformly amenable if there exists afunction F : N × (0 , → N such that each group Γ ∈ G is uniformly amenable with respect to F .See [31, §4] for properties of uniformly amenable groups. For example, finite products ofuniformly amenable groups, subgroups of a uniformly amenable group, and extension of auniformly amenable group by a uniformly amenable group are all uniformly amenable. Allsolvable groups are uniformly amenable [31, Lemma 5.10].B.3. Syndeticity.
We recall the definition of a syndetic subset in a group.
Definition B.2 (Syndeticity) . Let Γ be a discrete group. A subset E ⊂ Γ is said to be syndetic ifthere exists a finite set F in Γ such that FE ≔ S γ ∈ F γ E = Γ , in other words finitely many shiftsof E cover all of Γ .Here is a well-known test to determine when a set is syndetic. Proposition B.3.
Let Γ be a discrete amenable group. Let E be a subset of Γ . If for every leftFølner net ( Φ α ) α ∈ A for Γ there exists α ∈ A such that Φ α ∩ E , ∅ , then E is syndetic.Proof. Towards a contradiction assume that E is not syndetic. Let ( Φ α ) α ∈ A be an arbitrary Følnernet. Since we assumed that E is not syndetic and each Φ α is finite, we must have that thereexists h α ∈ Γ \ Φ − α E for each α ∈ A . Now ( Φ α h α ) α ∈ A is a Følner net (as can be easily seen fromtranslation invariance of Haar counting measure on Γ ) such that Φ α h α ∩ E = ∅ contradicting thehypothesis, thus E is syndetic. (cid:3) A suitable way to quantify syndeticity is the lower Banach density which we define next.
Definition B.4.
Let Γ be a discrete amenable group. For a subset E ⊂ Γ , we define the lowerBanach density of E to be the quantity BD Γ ( E ) ≔ inf { µ ( E ) : µ invariant finitely additive probability measure } . Bergelson, Hindman and McCutcheon established the following useful characterization ofsyndeticity in discrete amenable groups in [3, Theorem 2.7(a)].
Lemma B.5.
Let Γ be a discrete amenable group. A subset E ⊂ Γ is syndetic if and only if BD Γ ( E ) > . A ppendix C. U ltraproducts of measure - preserving dynamical systems An ultrafilter on N is a non-empty collection p of subsets of N satisfying the followingproperties:(i) ∅ < p ,(ii) A ∩ B ∈ p whenever A , B ∈ p ,(iii) B ∈ p whenever A ∈ p , A ⊂ B ,(iv) for all A ⊂ N either A ∈ p or A c ∈ p .Property (iv) distinguishes an ultrafilter from a filter. An ultrafilter p is said to be principal ifthere is a non-empty set A ⊂ N such that p = { B ∈ P ( N ) : A ⊂ B } . An ultrafilter is non-principal if it is not principal. The existence of non-principal ultrafilters is only guaranteed by the axiomof choice. More precisely, consider the Fréchet filter F which is the smallest filter containing allcofinal sets { n , n + , . . . } , n ∈ N . By the Boolean prime ideal theorem, there exists an ultrafilter p containing F . By construction, p is non-principal. On the other hand, any non-principalultrafilter p contains the Fréchet filter. Indeed, since p is an ultrafilter, it must contain either { n , n + , . . . } or its complement by property (iv) above. But if it contained the complement of { n , n + , . . . } , then it would be a principal ultrafilter.In what follows, the ultrafilter p on N is fixed. Let G be a uniformly amenable set of discretegroups. For each n ∈ N , let Γ n ∈ G and ( X n , µ n , T n ) be a PrbAlg op Γ n -dynamical system. For each n ∈ N , let ( Conc ( X n ) , B a ( Conc ( X n )) , µ Conc ( X n ) , T Conc ( X n ) ) be the corresponding canonical modelin CHPrb Γ n (see Definition 2.1). NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 27
The aim of this appendix is to sketch the construction of a
PrbAlg op Γ ∗ -dynamical system in-duced by a CncPrb Γ ∗ -dynamical system associated to an ultraproduct measure-preserving dy-namical system of the sequence of systems ( Conc ( X n ) , B a ( Conc ( X n )) , µ Conc ( X n ) , T Conc ( X n ) ), where Γ ∗ is the ultraproduct group of the sequence ( Γ n ) of uniformly amenable groups. The group Γ ∗ is defined as the quotient group of Q n ∈ N Γ n with respect to the equivalence relation ( γ n ) ∼ ( ˜ γ n )whenever { n ∈ N : γ n = ˜ γ n } ∈ p .The following characterization of uniform amenability was proved by Keller in [31, Theorem4.3 and Lemma 5.3]. Proposition C.1.
A discrete group Γ is uniformly amenable if and only if for every non-principalultrafilter p on N the ultrapower group Γ ∗ = Y n → p Γ is amenable (given the discrete topology). Similarly, a set G of discrete groups is uniformlyamenable if and only if for every non-principal ultrafilter p on N and for any sequence ( Γ n ) ofgroups in G the ultraproduct Γ ∗ = Y n → p Γ n is amenable (given the discrete topology). Similarly to Γ ∗ , we define the ultraproduct X ∗ = Q n → p Conc ( X n ) as the set of equivalenceclasses of elements of the product Q n ∈ N Conc ( X n ) with respect to the equivalence relation ( x n ) ∼ ( y n ) defined by { n ∈ N : x n = y n } ∈ p . Let A = { Y n → p E n : ( E n ) ∈ Y n ∈ N B a ( Conc ( X n )) } . Using the ultrafilter axioms, one can verify that A is an algebra of subsets of X ∗ .The Loeb premeasure µ ∗ : A → [0 ,
1] is defined by µ ∗ ( Y n → p E n ) : = st (lim n → p µ Conc ( X n ) ( E n )) , where st denotes the standard part of a non-standard real. Using the countable saturationproperty and Carathéodory’s extension theorem, µ ∗ can be extended to a measure µ on the σ -algebra X = σ ( A ) generated by A in X ∗ . Let ( X µ , ¯ µ ) denote the probability algebra of ( X , X , µ ).Let us denote by Aut( A , µ ∗ ) the automorphism group of ( A , µ ∗ ), that is the group of Booleanisomorphisms f : A → A such that µ ∗ ( E ) = µ ∗ ( f ( E )) for all E ∈ A . By chasing definitions,one can check that the sequence ( T Conc ( X n ) ) of measure-preserving continuous actions induces aconcrete action T ∗ : Γ ∗ → Aut( A , µ ∗ ) by defining( T ∗ ) γ ∗ ( Y n → p E n ) ≔ Y n → p T γ n Conc ( X n ) ( E n ) for all γ ∗ = [( γ n )] ∈ Γ ∗ and Q n → p E n ∈ A . By construction, A is an algebra of sets which isdense in X with respect to the pseudo-metric d ( E , F ) = µ ( E ∆ F ) on X . We define the abstractaction ¯ T : Γ ∗ → Aut( X µ , ¯ µ ) by ¯ T γ ∗ ([ E ]) ≔ _ n [ T γ ∗ ∗ ( E n )]where [ E ] denotes the equivalence class of E ∈ X in X µ , and ( E n ) is a sequence in A such that µ ( E n ∆ E ) → n tends to infinity. Observe that the definition of ¯ T γ ∗ ([ E ]) is independent ofthe choice of representatives and the approximating sequence. We thus obtain a PrbAlg op Γ ∗ -dynamical system ( X µ , ¯ µ, ¯ T ).Finally, suppose that for each n ∈ N , ( X n , µ n , S n ) is another PrbAlg op Γ n -dynamical systemsuch that S n and T n commute. Construct ( X µ , ¯ µ, ¯ S ) analogously to ( X µ , ¯ µ, ¯ T ) as before. Then S and T commute (which is easily seen by first verifying commutativity of S ∗ and T ∗ on A ), andtherefore ( X µ , ¯ µ, ¯ S , ¯ T ) becomes a PrbAlg op Γ ∗ -Roth dynamical system. Remark C.2.
In [12, §3,4], Conlon, Kechris and Tucker-Drob give an ultraproduct constructionof a sequence ( X n , X n , µ n , T n ) of CncPrb Γ -dynamical systems where ( X n , X n , µ n ) are standardBorel probability spaces and Γ is a fixed countably infinite group, which is based o ff a construc-tion of Elek and Szegedy [15] for finite probability spaces. They define a pointwise action of Γ (and not its ultrapower) on the Loeb probability space associated to the sequence ( X n , X n , µ n )by taking the ultralimit of the sequence ( T n ) of the pointwise actions.A ppendix D. A conditional H eine -B orel covering lemma This appendix is devoted to the following technical lemma which is needed in the proof ofTheorem 1.3. Throughout this section, suppose that Γ is some group and π : ( X , µ, T ) → ( Y , ν, S )a PrbAlg op Γ -factor map. We need the following notation. h f , g i X | Y : = E ( f ¯ g | Y ) , f , g ∈ L ( X ) , k f k X | Y : = E ( | f | | Y ) / , f ∈ L ( X ) , where ¯ g indicates complex conjugation. Lemma D.1.
Suppose
M ⊂ L ( X ) is a finitely generated, closed, and Γ -invariant L ∞ ( Y ) sub-module of L ( X ) . Let f ∈ M be such that kk f k X | Y k L ∞ ( Y ) < ∞ . Then for every ε > there exists afinite set N ⊂ M such that for every γ ∈ Γ , (D.1) min h ∈N k T γ ( f ) − h k X | Y ≤ ε. First we establish the following auxiliary result.
Proposition D.2 (A conditional Gram-Schmidt process) . Let M be a finitely generated andclosed L ∞ ( Y ) submodule of L ( X ) . Then there exist a partition ( E j ) j = ,..., m of Conc ( Y ) and afamily ( M j ) j = ,..., m of finite subsets of M satisfying the following properties. NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 29 (i) M = m X j = X u j ∈M j a u j u j E j : a u j ∈ L ∞ ( Y ) for all u j ∈ M j and j = , . . . , m . (ii) k u k X | Y = on E j for all u ∈ M j and j = , . . . , m. (iii) h u , u ′ i X | Y = on E j for all distinct u , u ′ ∈ M j and j = , . . . , m.Notice that (i), in particular, implies that M = { } on E .Proof. Since M is a finitely generated L ∞ ( Y ) submodule of L ( X ), there are f , . . . , f n ∈ L ( X )such that M = n X i = a i f i : a i ∈ L ∞ ( Y ) , i = , . . . , n . Let u = f k f k X | Y on {k f k X | Y > } , u ∈ M as 1 / k f k X | Y {k f k X | Y > } may not be in L ∞ ( Y ). For N ∈ N , let u N = f k f k X | Y on { / N ≤ k f k X | Y ≤ N } , u N converges to u almost surely as N tends to infinity and thus also in L ( X ) by dominatedconvergence. Since M is L closed we have showed u ∈ M .Next suppose we defined u , . . . , u k with k ≤ n −
1. Then set g k + = f k + − P ki = h f k + , u i i X | Y u i and define u k + = g k + k g k + k X | Y on {k g k + k X | Y > } , u , one can show that u k + is an element of M (where we now have to approximate first h f i , u i i X | Y in L ∞ ( Y ), then g k + and finally u k + ).Denote by F i = {k g i k X | Y > } and F i + n = {k g i k X | Y > } c for all i = , . . . , n . Form all finiteintersections F i ∩ F i ∩ . . . ∩ F i k with 1 ≤ i < i < . . . < i k ≤ n for some 1 ≤ k ≤ n . Let E denote the collection of such finite intersections whose measure is positive. Then E forms apartition of Conc ( Y ). Pick an element E = F i ∩ F i ∩ . . . ∩ F i k ∈ E and let M E = { u i t : i t ≤ n } . Let E denote the collection of elements E of E such that M E , ∅ . Now enumerate the elements of E by E , . . . , E m and correspondingly write M j = M E j for j = , . . . , m . Set E = ( S mj = E j ) c .By construction, ( E j ) j = ,..., m and ( M j ) j = ,..., m satisfy the desired properties (i), (ii), and (iii). (cid:3) We can prove our conditional Heine-Borel covering lemma.
Proof of Lemma D.1.
Suppose that
M ⊂ L ( X ) is a finitely generated, closed and Γ -invariant L ∞ submoldule of L ( X ). Let ( E j ) j = ,..., m and ( M j ) j = ,..., m be as in Proposition D.2. By assumption,for all γ ∈ Γ (D.2) k T γ ( f ) k X | Y = S γ ( k f k X | Y ) ≤ C for some constant C >
0. For each γ ∈ Γ , we have k T γ ( f ) k X | Y = m X j = X u j ∈M j | a γ u j , j | / E j for some a γ u j , j ∈ L ∞ ( Y ). By (D.2) we have (cid:16)P u j ∈M j | a γ u j , j | (cid:17) / ≤ C for all j and γ .Hence, for any fixed ε > ≤ j ≤ m finitely many vectors b , j , . . . , b k j , j ∈ L ∞ ( Y ) |M j | with b p , j = ( b , p , j , . . . , b |M j | , p , j ) for p = , . . . , k j such that a ∈ L ∞ ( Y ) |M j | : |M j | X q = | a q | / ≤ C ⊂ k j [ p = a ∈ L ∞ ( Y ) |M j | : |M j | X q = | a q − b q , p , j | / ≤ ε . Let N be the collection of all functions m X j = |M j | X q = b q , p j , j u j E j for some choice 1 ≤ p j ≤ k j for each j . Note that N is finite, and by construction N ⊂ M satisfies (D.1). (cid:3)
Remark D.3.
The results of this section are inspired by conditional analysis and conditionalset theory in [17, 9, 13]. The existence of a conditional orthonormal basis for certain L sub-modules of ( L ) d , d ≥ L denotes the algebra of equivalence classes of all complex measurable functions. Aconditional version of the Heine-Borel theorem within conditional set theory is established in[13, Theorem 4.6]. It is crucial in the conditional analysis of L modules to assume a closed-ness property under countable gluings which is referred to as σ -stability [9] or stability undercountable concatenations [17, 13].A main di ff erence in our analysis to the previously cited articles is that we work with L ∞ submodules of L spaces rather than with the larger L modules. However L ∞ modules donot satisfy this countable gluing property in general. We still manage to develop some usefulportion of conditional analysis for the smaller L ∞ modules by additionally requiring that thesemodules are finitely generated and closed in the L topology. These requirements are naturallysatisfied in the context of compact extensions in structural ergodic theory.R eferences [1] T. Austin. On the norm convergence of non-conventional ergodic averages. Ergodic Theory Dynam. Systems ,30(2):321–338, 2010.[2] T. Austin. Non-conventional ergodic averages for several commuting actions of an amenable group.
J. Anal.Math. , 130:243–274, 2016.[3] V. Bergelson, N. Hindman, and R. McCutcheon. Notions of size and combinatorial properties of quotient setsin semigroups. In
Proceedings of the 1998 Topology and Dynamics Conference (Fairfax, VA) , volume 23,pages 23–60, 1998.[4] V. Bergelson, B. Host, and B. Kra. Multiple recurrence and nilsequences.
Invent. Math. , 160(2):261–303,2005. With an appendix by Imre Ruzsa.
NIFORM SYNDETICITY OF MULTIPLE RECURRENCE TIMES 31 [5] V. Bergelson, B. Host, R. McCutcheon, and F. Parreau. Aspects of uniformity in recurrence.
Colloq. Math. ,84 / Topological dynamics and applications (Minneapolis, MN, 1995) , volume 215 of
Contemp. Math. , pages205–222. Amer. Math. Soc., Providence, RI, 1998.[7] V. Bergelson, R. McCutcheon, and Q. Zhang. A Roth Theorem for Amenable Groups.
Amer. J. of Math. ,119:1173–1211, 1997.[8] M. Boshernitzan, N. Frantzikinakis, and M. Wierdl. Under-recurrence in the Khintchine recurrence theorem.
Israel J. Math. , 222(2):815–840, 2017.[9] P. Cheridito, M. Kupper, and N. Vogelpoth. Conditional analysis on R d . Set Optimization and Applications,Proceedings in Mathematics & Statistics , 151:179–211, 2015.[10] Q. Chu. Multiple recurrence for two commuting transformations.
Ergodic Theory Dynam. Systems ,31(3):771–792, 2011.[11] Q. Chu and P. Zorin-Kranich. Lower bound in the Roth theorem for amenable groups.
Ergodic Theory Dynam.Systems , 35(6):1746–1766, 2015.[12] C. T. Conley, A. S. Kechris, and R. D. Tucker-Drob. Ultraproducts of measure preserving actions and graphcombinatorics.
Ergodic Theory Dynam. Systems , 33(2):334–374, 2013.[13] S. Drapeau, A. Jamneshan, M. Karliczek, and M. Kupper. The algebra of conditional sets and the concepts ofconditional topology and compactness.
J. Math. Anal. Appl. , 437(1):561–589, 2016.[14] T. Eisner, B. Farkas, M. Haase, and R. Nagel.
Operator theoretic aspects of ergodic theory , volume 272 of
Graduate Texts in Mathematics . Springer, Cham, 2015.[15] G. Elek and B. Szegedy. A measure-theoretic approach to the theory of dense hypergraphs.
Adv. Math. ,231(3-4):1731–1772, 2012.[16] R. Ellis. Topological dynamics and ergodic theory.
Ergodic Theory Dynam. Systems , 7(1):25–47, 1987.[17] D. Filipovi´c, M. Kupper, and N. Vogelpoth. Separation and duality in locally L -convex modules. J. Funct. Anal. , 256:3996–4029, 2009.[18] N. Frantzikinakis and R. McCutcheon. Ergodic theory: recurrence. In
Mathematics of complexity and dynam-ical systems. Vols. 1–3 , pages 357–368. Springer, New York, 2012.[19] D. H. Fremlin.
Measure theory. Vol. 3 . Torres Fremlin, Colchester, 2004. Measure algebras, Corrected secondprinting of the 2002 original.[20] H. Furstenberg. Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic progres-sions.
J. Anal. Math. , 31:204–256, 1977.[21] H. Furstenberg.
Recurrence in Ergodic Theory and Combinatorial Number Theory . Princeton Legacy Library.Princeton University Press, 2014.[22] H. Furstenberg and Y. Katznelson. An ergodic Szemerédi theorem for commuting transformations.
J. Anal.Math. , 34:275–291, 1978.[23] H. Furstenberg and Y. Katznelson. An ergodic Szemerédi theorem for IP-systems and combinatorial theory.
J. Analyse Math. , 45:117–168, 1985.[24] E. Glasner.
Ergodic Theory via Joinings . Mathematical Surveys and Monographs. American MathematicalSociety, 2015.[25] B. Host. Ergodic seminorms for commuting transformations and applications.
Studia Math. , 195(1):31–49,2009.[26] B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds.
Ann. Math. , 161(1):397–488, 2005.[27] A. Jamneshan. An uncountable Furstenberg-Zimmer structure theory.
In preperation , 2020.[28] A. Jamneshan and T. Tao. An uncountable Moore-Schmidt theorem. arXiv:1911.12033 , 2019.[29] A. Jamneshan and T. Tao. An uncountable Mackey-Zimmer theorem. arXiv:2010.00574 , 2020.[30] A. Jamneshan and T. Tao. Foundational aspects of uncountable measure theory: Gelfand duality, Riesz rep-resentation, canonical models, and canonical disintegration. arXiv:2010.00681 , 2020.[31] G. Keller. Amenable groups and varieties of groups.
Illinois J. Math. , 16:257–269, 1972.[32] A. Khintchine. Eine Verschärfung des Poincaréschen “Wiederkehrsatzes”.
Compositio Math. , 1:177–179,1935.[33] S. Koppelberg. General Theory of Boolean Algebras. In J. Bonk and R. Bonnet, editors,
Handbook of BooleanAlgebras . North-Holland, 1989. [34] A. F. Moragues. Properties of multicorrelation sequences and large returns under some ergodicity. arXiv:2006.03170 , 2020.[35] A. L. T. Paterson.
Amenability , volume 29 of
Mathematical Surveys and Monographs . American Mathemat-ical Society, Providence, RI, 1988.[36] J. Peterson. Lecture Notes on Ergodic Theory, 2011.[37] D. Robertson. Characteristic factors for commuting actions of amenable groups.
J. Anal. Math. , 129:165–196,2016.[38] K. F. Roth. Sur quelques ensembles d’entiers.
C.R. Acad. Sci. Paris 234 , pages 388–390, 1952.[39] R. J. Silverman. Means on semigroups and the Hahn-Banach extension property.
Trans. Amer. Math. Soc. ,83:222–237, 1956.[40] R. J. Silverman. Invariant means and cones with vector interiors.
Trans. Amer. Math. Soc. , 88:75–79, 1958.[41] E. Szemerédi. On sets of integers containing no k elements in arithmetic progression. Acta. Arith. , 27:199–245, 1975.[42] N. M. Walsh. Norm convergence of nilpotent ergodic averages.
Ann. of Math. , 175(3):1667–1688, 2012.[43] T. Ziegler. Universal characteristic factors and Furstenberg averages.
J. Amer. Math. Soc. , 20:53–97, 2007.[44] R. J. Zimmer. Ergodic actions with a generalized spectrum.
Illinois J. Math. , 20:555–588, 1976.[45] R. J. Zimmer. Extension of ergodic group actions.
Illinois J. Math. , 20:373–409, 1976.[46] P. Zorin-Kranich. Norm convergence of multiple ergodic averages on amenable groups.
J. Anal. Math. ,130:219–241, 2016.P olona D urcik , S chmid C ollege of S cience and T echnology , C hapman U niversity , O range , CA, USA Email address : [email protected] R achel G reenfeld , D epartment of M athematics , U niversity of C alifornia , L os A ngeles , CA, USA Email address : [email protected] A nnina I seli , D epartment of M athematics , U niversity of C alifornia , L os A ngeles , CA, USA Email address : [email protected] A sgar J amneshan , D epartment of M athematics , U niversity of C alifornia , L os A ngeles , CA, USA Email address : [email protected] J os ´ e M adrid , D epartment of M athematics , U niversity of C alifornia , L os A ngeles , CA, USA Email address ::