Complete type amalgamation and Roth's theorem on arithmetic progressions
aa r X i v : . [ m a t h . L O ] S e p COMPLETE TYPE AMALGAMATION AND ROTH’S THEOREMON ARITHMETIC PROGRESSIONS
AMADOR MARTIN-PIZARRO AND DANIEL PALACÍN
Abstract.
We extend previous work on Hrushovski’s stabilizer’s theorem andprove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers and ofNikolov-Pyber, on products of three sets and yields model-theoretic proofs ofexisting asymptotic results for quasirandom groups. Furthermore, we boundthe number of solutions to certain equations, such as x n · y m = z k for n + m = k ,in subsets of small tripling in groups. In particular, we show the existence oflower bounds on the number of arithmetic progressions of length for subsetsof small doubling without involutions in arbitrary abelian groups. Introduction
Szemerédi answered positively a question of Erdős and Turán by showing [28]that every subset A of N with upper density lim sup n →∞ | A ∩ { , . . . , n }| n > must contain an arithmetic progression of length k for every natural number k .For k = 3 , the existence of arithmetic progressions of length (in short -AP)was already proven by Roth in what is now called Roth’s theorem on arithmeticprogressions [22] (not to be confused with Roth’s theorem on diophantine approxi-mation of algebraic integers). There has been (and still is) impressive work done onunderstanding Roth’s and Szemerédi’s theorem, explicitly computing lower boundsfor the density as well as extending these results to more general settings. In thesecond direction, it is worth mentioning Green and Tao’s result on the existence ofarbitrarily long finite arithmetic progressions among the subset of prime numbers[7], which however has upper density .In the non-commutative setting, proving single instances of Szemerédi’s theorem,particularly Roth’s theorem, becomes highly non-trivial. Note that the sequence ( a, ab, ab ) can be seen as a -AP, even for non-commutative groups. Gowers asked[8, Question 6.5] whether the proportion of pairs ( a, b ) in PSL ( q ) , for q a primepower, such that a , ab and ab all lie in a fixed subset A of density δ approximatelyequals δ . For length , Gower’s question was positively answered by Tao [30] andlater extended to arbitrary non-abelian finite simple groups by Peluse [20]. For Date : September 21, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Model Theory, Additive Combinatorics, Arithmetic Progressions,Quasirandom Groups.Research supported by MTM2017-86777-P as well as by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) - Project number 2100310201 and 2100310301, part of theANR-DFG program GeoMod. arithmetic progressions ( a, ab, ab , ab ) of length in PSL ( q ) , a partial result wasobtained in [30], whenever the element b is diagonalizable over the finite field F q (which happens half of the time).A different generalization of Roth’s theorem, present in work of Sanders [23]and Henriot [9], is on the existence of a -AP in finite sets of small doubling inabelian groups. Recall that a finite set A of a group has doubling at most K ifthe productset A · A = { ab } a,b ∈ A has cardinality | A · A | ≤ K | A | . More generally,a finite set has tripling at most K if | A · A · A | ≤ K | A | . If A has tripling at most K , the comparable set A ∪ A − ∪ { id G } (of size at most | A | + 1 ) has triplingat most ( CK C ) with respect to some explicit absolute constant C > , so wemay assume that A is symmetric and contains the neutral element. Archetypalsets of small doubling are approximate subgroups, that is, symmetric sets A suchthat A · A is covered by finitely many translates of A . The model-theoretic studyof approximate subgroups first appeared in Hrushovski’s striking paper [11], whichcontained the so-called stabilizer theorem, adapting techniques from stability theoryto an abstract measure-theoretic setting. Hrushovski’s work has led to severalremarkable applications to additive combinatorics.In classical stability theory, and more generally, in a group G definable in a simpletheory, Hrushovski’s stabilizer of a generic type over an elementary substructure M is the connected component G M , that is, the smallest type-definable subgroupover M of bounded index (bounded with respect to the saturation of the ambientuniversal model). Types in G M are called principal types . If the theory is stable,there is a unique principal type, but this need not be the case for simple theories.However, Pillay, Scanlon and Wagner noticed [21, Proposition 2.2] that, given threeprincipal types p , q and r over M , there are independent realizations a of p and b of q over M such that ab realizes r . The main ingredient in their proof is a cleverapplication of -complete amalgamation (also known as the independence theorem)over the elementary substructure M . For the purpose of the present work, we shallnot define what a general complete amalgamation problem is, but a variation ofit, restricting the problem to conditions given by products with respect to theunderlying group law: Question.
Fix a natural number n ≥ . For each non-empty subset F of { , . . . , n } ,let p F be a principal type over the elementary substructure M . Can we find (undersuitable conditions) an independent (that is, weakly random) tuple ( a , . . . , a n ) of G n such that for all ∅ 6 = F ⊆ { , . . . , n } , the element a F realizes p F , where a F stands for the product of all a i , with i in F , written with the indices in increasingorder? The above formulation resonates with [7, Theorem 5.3] for quasirandom groupsand agrees for n = 2 with the aforementioned result of Pillay, Scanlon and Wagner.In this work, we will give a (partial) positive solution for n = 2 (Theorem 3.3) tothe above question for groups arising from ultraproducts of groups equipped withthe associated counting measure localized with respect to a distinguished finite set(Example 1.3). As a by-product, we obtain the corresponding version of the resultof Pillay, Scanlon and Wagner (Corollary 3.4): Theorem A.
Given a pseudo-finite subset X of small tripling in a sufficientlysaturated group G , for any three weakly random principal types p , q and r over a MALGAMATION AND ROTH’S THEOREM 3 countable elementary substructure in the subgroup generated by X there is a weaklyrandom pair ( a, b ) in p × q with a · b realizing r . This approach allows to unify both the existence of solutions to certain equationsin subsets of small tripling, as well as to reprove model-theoretically some of theknown results for ultra-quasirandom groups, that is, asymptotic limits of quasir-andom groups, already studied by Bergelson and Tao [2], and later by the secondauthor [19].A finite group is said to be d -quasirandom if all its non-trivial representationshave degree at least d ≥ . By a standard Łoś argument, we will show in Theorem4.8 the following result: Theorem B. ( cf. [8, Theorem 3.3 & Theorem 5.3]) Fix a natural number n ≥ .For every ∅ 6 = F ⊆ { , . . . , n } let δ F > be given. For every ǫ > there is someinteger d = d ( n, δ F , ǫ ) such that for every finite d -quasirandom group G and subsets A F of G of density at least δ F , the set X n = { ( a , . . . , a n ) ∈ G n | a F ∈ A F for all ∅ 6 = F ⊆ { , . . . , n }} has size |X n | ≥ − ǫ | G | n − − n Y F | A F | . In particular, for any three sets A , B and C of the d -quasirandom group G , |{ ( a, b, c ) ∈ A × B × C | ab = c }| > − ǫ | G | | A || B || C | . Setting q and r equal to p in Theorem A, we can easily deduce a finitary (albeitnon-quantitative) version of Roth’s theorem on -AP for finite subsets of smalldoubling in abelian groups with trivial -torsion (Corollary 5.3), which resemblesprevious work of Sanders [23, Theorem 7.1] and generalizes a result of Frankl,Graham and Rödl [5, Theorem 1]. Theorem C.
For every K ≥ , there exists some η = η ( K ) > such that, given anarbitrary abelian group G and a finite subset A of G of doubling at most K withoutelements of order , the set A contains at least η | A | many arithmetic progressionsof length . An arithmetic progression ( a, a + b, a + 2 b ) of length in an abelian group isequivalent to consider three elements x , y and z with x + z = 2 y (setting x = a , y = a + b and z = a + 2 b ). Thus, Roth’s Theorem is equivalent to the existence ofsolutions of the equation xz = y in abelian groups. Fre˘ıman [6] showed that thedoubling constant of finite subsets of the integers of small doubling containing no -AP’s must tend to infinity as the size of the set increases. Finer lower boundsfor the doubling constant in arbitrary abelian groups were obtained by Sanders in[23, Theorem 2.3]. In [24, Theorem 1.2] Sanders bounds quantitatively the size ofa subset A of a finite group G such that A contains no solutions to the equation xz = y with x = y . Our methods in the proof of Theorem C are not quantitative,yet they can be easily adapted to the study of solutions for other equations, forexample the equation x n y m = z r with n + m = r . We obtain in particular avariation of Theorem B, and hence of Roth’s Theorem (see Theorem 5.1): Theorem D.
For every constant K ≥ and natural numbers n , n and n with n + n = n , there is some λ = λ ( K, n , n , n ) > with the following property: In AMADOR MARTIN-PIZARRO AND DANIEL PALACÍN an arbitrary group, given a finite subset A of tripling at most K without non-trivialelements of order dividing n , n or n , |{ ( x, y, z ) ∈ A × A × A | x n · y n = z n }| ≥ λ | A | . The reader will easily remark that Theorem D implies Theorem C, setting n =1 = n and n = 2 , and switching the role of y and z .Another relevant equation we may consider is xy = z . Indeed, a monochromaticsolution to the equation in a group G equipped with a finite coloring represents amonochromatic triangle. Schur’s theorem [26] asserts the existence of monochro-matic triangles for any finite coloring of the natural numbers , . . . , N , whenever N is sufficiently large. Sanders noticed [25, Theorem 1.1] that Schur’s proof couldbe adapted to determine the number of monochromatic triangles. Our techniquesyield a non-quantitative proof of Sander’s result (see Theorem 5.4): Theorem E.
For every natural number k , there is some ν = ν ( k ) > with thefollowing property: Given any coloring on a finite group G with k many colors A , . . . , A k , there exists some color A j , with ≤ j ≤ k , such that |{ ( a, b, c ) ∈ A j × A j × A j | a · b = c }| ≥ ν | G | . We finish this introduction with a couple of remarks on the structure and pre-sentation of this article: whilst almost all of the statements presented so far are ofcombinatorial nature, our proofs are model-theoretic. Hence, we assume through-out the text a certain familiarity with basic notions in model theory. Sections , and contain the model-theoretic core of the paper, whilst Sections 4 and 5 con-tain applications to additive combinatorics. In Section 4, we revisit the notion ofquasirandom groups and reprove using our techniques some of the results of [8] and[2]. We also include in Section 4 an aside containing a local approach weakeningthe notion of quasi-randomness to given sets. Finally, in Section 5, we concentrateon solutions of equations in groups.1. Randomness and Fubini
Most of the material in this section can be found in [11, 16].We work inside a sufficiently saturated model U of a complete first-order theory(with infinite models) in a language L , that is, the model U is saturated and stronglyhomogeneous with respect to some sufficiently large cardinal κ . All sets and tuplesare taken inside U .A subset X of U n is definable over the parameter set A if there exists a formula φ ( x , . . . , x n , y , . . . , y m ) and a tuple a = ( a , . . . , a m ) in A such that an n -tuple b belongs to X if and only if φ ( b, a ) holds in U . As usual, we identify a definablesubset of U with a formula defining it. Unless explicitly stated, when we use theword definable, we mean definably possibly with parameters. It follows that a subset X is definable over the parameter set A if and only if X is definable (over someset of parameters) and invariant under the action of the group of automorphisms Aut( U /A ) of U fixing A pointwise. The subset X of U is type-definable if it is theintersection of a bounded number of definable sets, where bounded means that itssize is strictly smaller than the degree of saturation of U .For the applications we will mainly consider the case where the language L contains the language of groups and the universe of our ambient model is a group. MALGAMATION AND ROTH’S THEOREM 5
Nonetheless, our model-theoretic setting works as well for an arbitrary definablegroup, that is, a group whose underlying set and its group law are both definable.
Definition 1.1. A definably amenable pair ( G, X ) consists of a definable group G together with a definable subset X of G such that there is a finitely additive measure µ on the definable subsets on the subgroup h X i generated by X with µ ( X ) = 1 and which is in addition invariant under left and right translation.Note that the subgroup h X i need not be definable, but it is locally definable , forthe subgroup h X i is a countable union of definable sets of the form X ⊙ n = X · · · X | {z } n , where X is the definable set X ∪ X − ∪ { id G } . Furthermore, every definable subset Y of h X i is contained in some finite product X ⊙ n , by compactness and saturationof the ambient model.Throughout the paper, we will always assume that the language L is rich enough(see [27, Definition 3.19]) to render the measure µ definable without parameters. Definition 1.2.
The measure µ of a definably amenable pair ( G, X ) is definablewithout parameters if for every L -formula ϕ ( x, y ) , every natural number n ≥ andevery ǫ > , there is a partition of the L -definable set { y ∈ U n | ϕ ( U , y ) ⊆ X ⊙ n } into L -formulae ρ ( y ) , . . . , ρ m ( y ) such that whenever a pair ( b, b ′ ) in U n × U n realizes ρ i ( y ) ∧ ρ i ( z ) , then | µ ( ϕ ( x, b )) − µ ( ϕ ( x, b ′ )) | < ǫ. The above definition is a mere formulation of [27, Definition 3.19] to the locallydefinable context, by imposing that the restriction of µ to every definable subset X ⊙ n is definable in the sense of [27, Definition 3.19]. In particular, a definable mea-sure of a definably amenable pair ( G, X ) is invariant , that is, its value is invariantunder the action of Aut( U ) . Example 1.3.
Let ( G n ) n ∈ N be an infinite family of groups, each with a distin-guished finite subset X n . Expand the language of groups to a language L includinga unary predicate and set M n to be an L -structure with universe G n , equippedwith its group operation, and interpret the predicate as X n . Following [11, Section2.6] we can further assume that L has predicates Q r,ϕ ( y ) for each r in Q ≥ andevery formula ϕ ( x, y ) in L such that Q r,ϕ ( b ) holds if and only if the set ϕ ( M n , b ) is finite with | ϕ ( M n , b ) | ≤ r | X n | . Note that if the original language was countable,so is the extension L .Consider now the ultraproduct M of the L -structures ( M n ) n ∈ N with respect tosome non-principal ultrafilter U . Denote by G and X the corresponding interpreta-tions in a sufficiently saturated elementary extension U of M . For each L -formula ϕ ( x, y ) and every tuple b in U | y | such that ϕ ( U , b ) is a subset of h X i , define µ ( ϕ ( x, b )) = inf (cid:8) r ∈ Q ≥ | Q r,ϕ ( b ) holds (cid:9) , where we assign ∞ if Q r,ϕ ( b ) holds for no value r . This is easily seen to be afinitely additive definable measure on the Boolean algebra of definable subsets of h X i , which is invariant under left and right translation. In particular, the pair ( G, X ) is definably amenable.We will throughout this paper consider two main examples: AMADOR MARTIN-PIZARRO AND DANIEL PALACÍN (a) The set X equals G itself, which happens whenever the subset X n = G n for U -almost all n in N . The normalized counting measure µ defined above is adefinable Keisler measure [13] on the pseudo-finite group G .(b) For U -almost all n , the set X n has small tripling : there is a constant K > such that | X n X n X n | ≤ K | X n | (or more generally | X n X − n X n | ≤ K | X n | ).The non-commutative Plünnecke-Ruzsa inequality [29, Lemma 3.4] yields that | X ⊙ mn | ≤ K O m (1) | X n | , so the measure µ ( Y ) is finite for every definable subset Y of h X i , since Y is then contained in X ⊙ m for some m in N . In particular,the measure µ is σ -finite as well.Whilst each subset X n in the example (b) must be finite, we do not impose thatthe groups G n are finite. If the set X n has tripling at most K , the set X ⊙ = X n ∪ X − n ∪ { id G } has size at most | X n | + 1 and tripling at most ( CK C ) withrespect to some explicit absolute constant C > . Thus, taking ultraproducts, bothstructures ( G, X ) and ( G, X ⊙ ) will have the same sets of positive measure (ordensity), though the values may differ. Hence, we may assume that, in a definablyamenable pair ( G, X ) , the corresponding definable set X is symmetric and containsthe neutral element of G .The construction in Example 1.3 can also be carried out for a finite cartesianproduct to produce for every n ≥ in N a definably amenable pair ( G n , X n ) , where h X n i = h X i n , equipped with a definable σ -finite measure µ n . Thus, the followingassumption is satisfied by our two main examples. Assumption 1.
For every n ≥ , the pair ( G n , X n ) is definably amenable for thedefinable σ -finite measure µ n .Carathéodory’s extension theorem implies the existence of a unique σ -additivemeasure on the σ -algebra generated by the definable subsets of h X i . We will denotethe extension again by µ n , though there will be (most likely) sets of infinite measure,as noticed by Massicot and Wagner: Fact 1.4. ([16, Remark 4]) The subgroup h X i is definable if and only if µ ( h X i ) isfinite.The extension of µ n to the σ -algebra generated by the definable subsets of h X i n isagain invariant under left and right translations, as well as under automorphisms:Indeed, every automorphism τ of Aut( U ) gives rise to a measure µ τn , such that µ τn ( Y ) = µ n ( τ ( Y )) for every measurable subset Y of h X i n . Since µ τn agrees with µ n on the collection of definable subsets, we conclude that µ τn = µ n by the uniquenessof the extension. Thus, the measure of a Borel subset Y in the space of typescontaining a fixed clopen set [ Z ] , where Z is a definable subset of h X i n , dependssolely on the type of the parameters defining Y .The definability condition in Definition 1.2 implies that the function S m ( C ) → R tp( b/C ) µ n ( ϕ ( x, b )) is well-defined and continuous for every L C -formula ϕ ( x, y ) with | x | = n and | y | = m such that ϕ ( x, y ) defines a subset of h X i n + m . Therefore, for such L C -formulae ϕ ( x, y ) , we can consider the following measure ν on h X i n + m , ν ( ϕ ( x, y )) = Z h X i m µ n ( ϕ ( x, y )) dµ m , MALGAMATION AND ROTH’S THEOREM 7 where the integral in fact runs over the L C -definable subset { y ∈ h X i m | ∃ x ϕ ( x, y ) } .For the pseudo-finite measures described in Example 1.3, the above integral equalsthe ultralimit lim k →U | X k | m X y ∈h X k i m | ϕ ( x, y ) || X k | n , so ν equals µ n + m and consequently Fubini-Tonelli holds. For arbitrary definablyamenable pairs, whilst the measure ν extends the product measure µ n × µ m , itneed not be a priori µ n + m [27, Remark 3.28]. Keisler [13, Theorem 6.15] exhibiteda Fubini-Tonelli type theorem for general Keisler measures under certain condi-tions. We will impose a further restriction on the definably amenable pairs we willconsider, taking Example 1.3 as a guideline. Assumption 2.
For every definably amenable pair ( G, X ) and its correspondingfamily of definable measures ( µ n ) n ∈ N on the Cartesian powers of h X i , the Fubinicondition holds: Whenever a definable subset of h X i n + m is given by an L C -formula ϕ ( x, y ) with | x | = n and | y | = m , the following equality holds: µ n + m ( ϕ ( x, y )) = Z h X i m µ n ( ϕ ( x, y )) dµ m = Z h X i n µ m ( ϕ ( x, y )) dµ n . Whilst this assumption is stated for definable sets, it extends to certain Borelsets, whenever the language L C is countable. Note indeed that for every Borelset Z ( x, y ) with | x | = n and | y | = m such that Z ( x, y ) is contained in a definablesubset of h X i n + m , definability and regularity of the measures yield that the function y µ ( Z ( x, y )) is Borel, thus measurable. Remark 1.5.
Assume that L C is countable and fix a natural number k ≥ . Forevery Borel set Z ( x, y ) with | x | = n and | y | = m contained as a subset in ( X ⊙ k ) n + m ,we have the identity µ n + m ( Z ( x, y )) = Z h X i m µ n ( Z ( x, y )) dµ m = Z h X i n µ m ( Z ( x, y )) dµ n , by a straightforward application of the monotone class theorem, as in [2, Theorem20], using the fact that µ ( X ⊙ k ) is finite. In particular, the identity holds for everyBorel set of finite measure by regularity. Henceforth, the language is countable and all definably amenable pairssatisfy Assumptions 1 and 2.
Adopting the terminology from additive combinatorics, we shall use the word density for the value of the measure of a subset in a definably amenable pair ( G, X ) .A (partial) type is said to be weakly random if it contains a definable subsetof positive density but no definable subset of density . Note that every weaklyrandom partial type Σ( x ) over a parameter set A can be completed to a weaklyrandom complete type over any arbitrary set B containing A , since the collectionof formulae Σ( x ) ∪ {¬ ϕ ( x ) | ϕ ( x ) L B -formula of density } is finitely consistent. Thus, weakly random types exist (yet the partial type x = x is not weakly random whenever G = h X i ). As usual, we say that an element b of G is weakly random over A if tp( b/A ) is.Weakly random elements satisfy a weak notion of transitivity. AMADOR MARTIN-PIZARRO AND DANIEL PALACÍN
Lemma 1.6.
Let b be weakly random over a set of parameters C and a be weaklyrandom over C, b . The pair ( a, b ) is weakly random over C .Proof. We need to show that every C -definable subset Z of h X i n + m containing thepair ( a, b ) has positive density with respect to the product measure µ n + m , where n = | a | and m = | b | . Since a is weakly random over C, b , the fiber Z b of Z over b has measure µ n ( Z b ) ≥ r for some rational number r > . Hence b belongs to the C -definable subset Y = { y ∈ U m | µ n ( Z y ) ≥ r } of h X i m , so µ m ( Y ) > . Thus, µ n + m ( Z ) = Z h X i m µ n ( Z y ) dµ m ≥ Z Y µ n ( Z y ) dµ m ≥ µ m ( Y ) r > , as desired. (cid:3) Note that the tuple b above may not be weakly random over C, a . To remedy thefailure of symmetry in the notion of randomness, we will introduce random types,which will play a fundamental role in Section 3. Random types already appear in[12, Exercise 2.25], so we solely recall Hrushovski’s definition of ω -randomness . Definition 1.7.
We define inductively on n in N the Boolean algebra Def n ( C ) ofsets of higher measurable complexity over a countable subset of parameters C : Thecollection Def ( C ) consists of the L C -definable subsets of h X i , whereas Def n +1 ( C ) is the Boolean algebra generated by both Def n ( C ) and all the sets of the form { a ∈ h X i k | µ m ( Z a ) = 0 } , where Z ⊆ h X i k + m runs over all subsets of Def n ( C ) .Note that the every subset in Def n ( C ) is Borel, so we can talk about their valuewith respect to the extensions of our original collection of σ -finite measures µ k .However, the algebra Def ( C ) contains new sets which are neither type-definablenor their complement is. Definition 1.8.
A tuple is random over the countable set C if it lies in no subset Z of Def n ( C ) of measure , for n in N .Randomness is a property of the type: If a and b have the same type over C ,then a is random over C if and only if b is. Note that if the tuple a of h X i is randomover C , then it is in particular weakly random over C , which justifies our choice ofterminology (instead of wide types).Notice that all the Boolean algebras Def n ( C ) are countable. Hence, since thevalue of the measure and its extension coincide for subsets of Def ( C ) , it followsby σ -additivity of the measure that no subset of Def ( C ) of positive measure canbe covered by Borel subsets of measure from higher Def n ’s, allowing to concludethe following result: Remark 1.9.
Every definable subset of h X i over the countable set C (that is, asubset in Def ( C ) ) of positive density contains a random element over C .Randomness is a symmetric notion. Lemma 1.10. ([12, Exercise 2.25])
A finite tuple ( a, b ) in h X i is random over C if and only if a is random over C and b is random over C, a .Proof.
Fix some natural number k ≥ such that the every coordinate of the tuple ( a, b ) belongs to X ⊙ k . Assume that ( a, b ) is random over C . Clearly so is a , thus MALGAMATION AND ROTH’S THEOREM 9 we need only prove that b is random over C, a . Suppose on the contrary that thereis a subset Z a of Def n ( C, a ) , for some n in N , of density containing b . Write Z ( a, y ) = Z a for some subset Z of h X i | a | + | b | in Def n ( C ) . Thus, the pair ( a, b ) belongs to ˜ Z = Z ∩ { ( x, y ) ∈ ( X ⊙ k ) | a | + | b | | Z x has density } , which is a subset in Def n +1 ( C ) , and thus it cannot have density . However,Remark 1.5 yields < µ | a | + | b | ( ˜ Z ) = Z h X i | a | µ | b | ( ˜ Z x ) dµ | a | = 0 , which gives the desired contradiction.Assume now that a is random over C and b is random over C, a . A verbatimtranslation (switching the roles of a and b ) of the proof of Lemma 1.6, using Remark1.5, yields that whenever ( a, b ) lies in a finite density subset Z of Def n ( C ) , then Z has positive measure. (cid:3) Symmetry of randomness will play an essential role in Section 3 allowing us totransfer ideas from the study of definable groups in simple theories to the pseudo-finite context. 2.
Forking and measures
As in Section 1, we work inside a sufficiently saturated structure and a definablyamenable pair ( G, X ) in a fixed countable language L satisfying Assumptions 1 and2, though the classical notions of forking and stability do not require the presenceof a group nor of a measure.Recall that a definable set ϕ ( x, a ) divides over a subset C of parameters if thereexists an indiscernible sequence ( a i ) i ∈ N over C with a = a such that the intersection T i ϕ ( x, a i ) is empty. Archetypal examples of dividing formulae are of the form x = a for some element a not algebraic over C . Since dividing formulae need not be closedunder disjunction, witnessed for example by a circular order, we say that a fomula ψ ( x ) forks over C if it belongs to the ideal generated by formulae dividing over C ,that is, if ψ implies a finite disjunction of formulae, each dividing over C . A type divides , resp. forks over C , if it contains an instance which does. Remark 2.1.
Since the measure is invariant under automorphisms and σ -finite,no definable subset of h X i of positive density can divide, thus a weakly randomtype does not fork over the empty-set.Non-forking need not define a tame notion of independence, for example it neednot be symmetric, yet it behaves extremely well with respect to certain invariantrelations, called stable. Definition 2.2.
An invariant relation R ( x, y ) is stable if there is no indiscerniblesequence ( a i , b i ) i ∈ N such that R ( a i , b j ) holds if and only if i < j. A straight-forward Ramsey argument yields that the collection of invariant stablerelations is closed under Boolean combinations. Furthermore, an invariant relation(without parameters) is stable if there is no indiscernible sequence as in the defini-tion of length some fixed infinite ordinal.The following remark will be very useful in the following sections.
Remark 2.3. ([11, Lemma 2.3]) Suppose that the type tp( a/M, b ) does not forkover the elementary substructure M and that the M -invariant relation R ( x, y ) isstable. Then the following are equivalent:(a) The relation R ( a, b ) holds.(b) The relation R ( a ′ , b ) holds, whenever a ′ ≡ M a and tp( a ′ /M b ) does not fork.(c) The relation R ( a ′ , b ) holds, whenever a ′ ≡ M a and tp( b/M a ′ ) does not fork.A clever use of the Krein-Milman theorem on the locally compact Hausdorff topo-logical real vector space of all σ -additive probability measures allowed Hrushovskito prove the following striking result: Proposition 2.4. ([11, Proposition 2.25])
Given a real number α and L M -formulae ϕ ( x, z ) and ψ ( y, z ) with parameters over an elementary substructure M , the M -invariant relation on the definably amenable pair ( G, X ) R αϕ,ψ ( a, b ) ⇔ µ | z | (cid:0) ϕ ( a, z ) ∧ ψ ( b, z ) (cid:1) = α is stable. In particular, for any partial types Φ( x, z ) and Ψ( y, z ) over M , the relation Q Φ , Ψ ( a, b ) ⇔ Φ( a, z ) ∧ Ψ( b, z ) is weakly randomis stable (cf. [11, Lemma 2.10] ). Strictly speaking, Hrushovski’s result in its original version is stated for arbitraryKeisler measures (in any theory). To deduce the statement above it suffices tonormalize the measure µ | z | by µ | z | (( X | z | ) ⊙ k ) , for a natural number k such that ( X | z | ) ⊙ k contains the corresponding instances of ϕ ( x, z ) and ψ ( y, z ) .We will finish this section with a summarized version of Hrushovski’s stabilizertheorem tailored to the context of definably amenable pairs. Before stating it, wefirst need to introduce some notation. Definition 2.5.
Let X be a definable subset of a definable group G and let M bean elementary substructure. We denote by h X i M the intersection of all subgroupsof h X i type-definable over M and of bounded index.If a subgroup of bounded index type-definable over M exists, the subgroup h X i M is again type-definable over M and has bounded index, see [11, Lemmata 3.2 & 3.3].Furthermore, it is also normal in h X i (cf. [11, Lemma 3.4]), since it is the kernel ofthe group homomorphism h X i → Sym( h X i / h X i M ) g σ g where σ g is the permutation mapping h h X i M → gh h X i M . Fact 2.6. ([11, Theorem 3.5] & [17, Theorem 2.12]) Let ( G, X ) be a definablyamenable pair and let M be an elementary substructure. For any weakly randomtype p over M contained in h X i , the subgroup h X i M exists and equals h X i M = ( p · p − ) , where we identify a type with its realizations in the ambient structure U . Fur-thermore, the set pp − p is a coset of h X i M . For every element a in h X i M weaklyrandom over M , the partial type p ∩ a · p is weakly random. MALGAMATION AND ROTH’S THEOREM 11
If the definably amenable pair we consider happens to be as in the first case ofExample 1.3, note that our notation coincides with the classical notation G M .Note that each coset of h X i M is type-definable over M and hence M -invariant,though it need not have a representative in M . Thus, every type p over M containedin h X i must determine a coset of h X i M . We denote by C M ( p ) the coset of h X i M of h X i containing some, and hence every, realization of p .3. On -amalgamation and solutions of xy = z As in Section 1, we fix a definably amenable pair ( G, X ) satisfying Assumption1 and 2, and work over some elementary countable substructure M . We denote by S M ( µ ) the support of µ , that is, the collection of all weakly random types over M contained in h X i . Lemma 3.1.
Given M -definable subsets A and B of h X i of positive density, thereexist some random element g over M with µ ( Ag ∩ B ) > .Proof. By Remark 1.9, let c be random in B over M and choose now g − in c − A random over M, c . The element g is also random over M, c . By symmetry ofrandomness, the pair ( c, g ) is random over M , so c is random over M, g . Clearlythe element c lies in Ag ∩ B , so the set Ag ∩ B has positive density, as desired. (cid:3) Remark 3.2.
Notice that the above results yields the existence of an element h random over M such that hA ∩ B , and thus A ∩ h − B , has positive density: Indeed,apply the statement to the definable subsets B − and A − .The next result was first observed for principal generic types in a simple theoryin [21, Proposition 2.2] and later generalized to non-principal types in [15, Lemma2.3]. For weakly random types with respect to a pseudo-finite Keisler measure, apreliminary (weaker) version was obtained by the second author [19, Proposition3.2] for ultra-quasirandom groups, which will be discussed in more detail in Section4. Theorem 3.3.
For any three types p , q and r in the support S M ( µ ) over M , thereare realizations a of p and b of q with a weakly random over M, b and a · b realizing r if and only if their cosets over M satisfy that C M ( p ) · C M ( q ) = C M ( r ) .Proof. Clearly, we need only prove the existence of the realizations a , b and c as inthe statement, provided that the cosets of p , q and r satisfy C M ( p ) · C M ( q ) = C M ( r ) .We proceed by proving the following auxiliary claims. Claim 1.
Given finitely many subsets A , . . . , A n in p and B , . . . , B n in r , thereexists a random element g in h X i over M with A i g ∩ B j of positive density for all ≤ i, j ≤ n .Proof of Claim 1. The definable subsets A = T ≤ i ≤ n A i and B = T ≤ i ≤ n B i lie in p and r respectively, hence they have positive density. Lemma 3.1 applied to A and B yields the desired random element g . (cid:3) Claim 1
Claim 2.
There exists some element g in h X i such that the partial type p · g ∩ r isweakly random. Proof of Claim 2.
Set Y = X ⊙ m for some natural number m such that the sym-metric set X ⊙ m contains all realizations of p and r . Working in the Stone space ofthe Boolean algebra Def ( M ) , the clopen set [ Y ] cannot be written as [ A ∈ pB ∈ r [ { x ∈ Y | µ ( Ax ∩ B ) = 0 } ] . Indeed, by compactness (of the Stone space of
Def ( M ) ), it suffices to show that [ Y ] cannot be covered by a finite union as above. Given A , . . . , A n in p and B , . . . , B n in r , which we may assume to be subsets of X ⊙ m , we find by Claim 1 an element g in Y and some δ > such that µ ( A i g ∩ B j ) ≥ δ for all ≤ i, j ≤ n . In particular,the L -type tp( g/M ) belongs to the clopen set [ { x ∈ Y | µ ( A i x ∩ B j ) ≥ δ for all ≤ i, j ≤ n } ] . Hence, no extension of tp( g/M ) (to an ultrafilter in the Stone space of Def ( M ) )lies in the finite union [ ≤ i,j ≤ n [ { x ∈ Y | µ ( A i x ∩ B j ) = 0 } ] . Choose therefore an element V of the Stone space of Def ( M ) lying in [ Y ] \ [ A ∈ pB ∈ r [ { x ∈ Y | µ ( Ax ∩ B ) = 0 } ] . For each A in p and B in r , the ultrafilter V must contain the set { x ∈ Y | µ ( Ax ∩ B ) > } , so V must contain, for some rational number δ > , the Def ( M ) -clopen set [ { x ∈ Y | µ ( Ax ∩ B ) ≥ δ } . Thus, the restriction of the above ultrafilter to Def ( M ) yields an L -type over M such that for each of its realization g in U , the partial type p · g ∩ r is weakly random, as desired. (cid:3) Claim 2
Since C M ( r ) = C M ( p ) · C M ( q ) , observe that any element g as in Claim 2 lies in C M ( q ) . Fix now such an element g and choose a realization b of q weakly randomover M, g . Since weakly random types do not fork, note that tp( bg − /M, g ) doesnot fork over M . Claim 3.
For some g in h X i weakly random over M, g, b , the type p · ( bg − g ) ∩ r is weakly random. In particular the type tp( g /M, b, g ) does not fork over M .Proof of Claim 3. Since s = tp( g/M ) lies in C M ( q ) , the difference bg − is a weaklyrandom element in the normal subgroup h X i M . Hence, the partial type s ∩ bg − s isweakly random over M, bg − by Fact 2.6. Choose an element g realizing s weaklyrandom over M, g, b such that bg − g ≡ M g as well. By invariance of the measure,we have that p · ( bg − g ) ∩ r is weakly random, as desired. (cid:3) Claim 3
Summarizing, the relation Q p,r ( u, v ) ⇔ “ p · ( u · v ) ∩ r is weakly random”holds for the pair ( bg − , g ) with tp( g /M, bg − ) non-forking over M . Note that theabove relation is stable, by Proposition 2.4, so Q p,r must hold for any pair ( w, z ) such that w ≡ M bg − , z ≡ M g & tp( w/M, z ) non-forking over M, MALGAMATION AND ROTH’S THEOREM 13 by the Remark 2.3. Setting w = bg − and z = g , we conclude that p · b ∩ r = p · ( bg − g ) ∩ r is weakly random over M . Choose now a realization c of p · b ∩ r weakly randomover M, b and set a = cb − , which realizes a weakly random extension of p to M, b by our choice of c . (cid:3) Corollary 3.4.
Given three weakly random types p , q and r in h X i M , the partialtype { ( x, y ) ∈ p × q | xy ∈ r } is weakly random in the definably amenable pair ( G , X ) .Proof. Since the above partial type is type-definable over M , it suffices to showthat it is realized by a weakly random pair over M . Choose by Theorem 3.3 a pair ( a, b ) realizing p × q with ab realizing r and such that a is weakly random over M, b .Thus, the tuple b is also weakly random over M and hence, so is the pair ( a, b ) byLemma 1.6. (cid:3) It follows from Lemma 3.1 that, for any two definable subsets A and B of positivedensity, there exists an element g in h X i such that the intersection A ∩ gB haspositive density as well. We will now see that this density is constant within acoset of h X i M . Corollary 3.5.
Given two subsets A and B of positive density definable over M ,the values µ ( A ∩ gB ) and µ ( A ∩ hB ) agree for any two weakly random elements g and h over M within the same coset of h X i M .Proof. Without loss of generality, it suffices to consider the case where the value µ ( A ∩ gB ) = α > for some weakly random element g over M and denote by r its type over M . Choose some weakly random type p in h X i M over M . Byconstruction C M ( r ) = C M ( p ) · C M ( r ) . Theorem 3.3 yields that g = cd for some realization d of r and some weakly randomelement c over M, d realizing p . By invariance of the measure, we still have that α = µ ( c − A ∩ dB ) .For any weakly random type s = tp( h/M ) in C M ( r ) , we clearly have that C M ( s ) = C M ( r ) , so Theorem 3.3 yields that h = c d for some realizations c of p and d of r with tp( c /M, d ) weakly random (thus non-forking over M ). Asthe relation R αA,B ( u, v ) ⇔ “ µ ( u − A ∩ vB ) = α ”is stable by Proposition 2.4, we conclude by the Remark 2.3 that µ ( A ∩ hB ) = α ,as desired. (cid:3) Ultra-quasirandom groups revisited
We begin this section by recalling the notion of quasirandomness introduced byGowers [8].
Definition 4.1.
Let d ≥ . A finite group is d -quasirandom if all its non-trivialrepresentations have degree at least d .To study the asymptotic behaviour of increasingly finite quasirandom groups,we shall consider ultraproducts, following Bergelson and Tao [2]. Definition 4.2.
An ultraproduct of finite groups ( G n ) n ∈ N with respect to a non-principal ultrafilter U is said to be ultra-quasirandom if for every d ≥ , the set { n ∈ N | G n is d -quasirandom } belongs to U .A sufficiently saturated extension of an ultra-quasirandom group need not be anultraproduct of finite groups (by cardinality reasons). We will nevertheless referto the saturated extension again as an ultra-quasirandom group in an abuse ofterminology justified by the following observation: Remark 4.3.
An ultra-quasirandom group M = Q U G n gives rise to a definablyamenable pair ( G, G ) with respect the normalized counting measure µ which sat-isfies Assumption 1 and 2, as discussed in Example 1.3(a). Furthermore, the workof Gowers [8, Theorem 3.3] yields that every definable subset A of the ultraprod-uct G ( M ) of positive density is not product-free , i.e. it contains a solution to theequation xy = z , and thus the same holds in any elementary extension. Therefore,definability of the measure µ yields that G = G N over any elementary substruc-ture N [14, Corollary 2.6]. As shown in [19, Theorem 4.8], the identity G = G M characterises (saturated extensions of) ultra-quasirandom groups.Throughout the section, we work in the setting of Example 1.3(a) with µ denotingthe normalized counting measure in the ultra-quasirandom group G (see Remark4.3).Theorem 3.3 and its corollaries yield now a shorter proof of (some of the equiv-alences in) [19, Theorem 4.8], which we include for the sake of completeness. Corollary 4.4.
Given three subsets A , B and C of positive density of an ultra-quasirandom group G , we have that G = A · B · C and the measure µ ( G \ AB − ) = 0 .Proof. Given three subsets A , B and C of positive density definable over somecountable elementary substructure M , we need only show that every element g in G ( M ) lies in A · B · C , which follows immediately from Corollary 3.4 by choosingweakly random types p in A , q in B and r in gC − over M .If the M -definable subset G \ AB − had positive density, we could find a weaklyrandom type r over M containing this set. Any choice of weakly random types p in A and q in B over M gives the desired contradiction by Theorem 3.3, since G M equals G . (cid:3) The following result on weak mixing, already present as is in the work of Taoand Bergelson, was implicit in the work of Gowers [8]. It will play a crucial role tostudy some instances of complete amalgamation for solving equations in a group.
Corollary 4.5. ( cf. [2, Lemma 33]) Given two subsets A and B of positive densitydefinable in an ultra-quasirandom group G , the measure µ ( A ∩ gB ) = µ ( A ) µ ( B ) for µ -almost all elements g .Proof. As before, fix some countable elementary substructure M such that both A and B are M -definable. Note that the measure µ is also definable over M .By Corollary 3.5, let α be the value of µ ( A ∩ gB ) for some (or equivalently, every)weakly random element g over M . Notice that α > by the Remark 3.2.In particular, the subset Z = { x ∈ AB − | µ ( A ∩ xB ) = α } MALGAMATION AND ROTH’S THEOREM 15 is type-definable over M and contains all weakly random elements over M , so µ ( Z ) = µ ( AB − ) = 1 , by Corollary 4.4.If we denote by µ the normalized counting measure in G , an easy computationyields that µ ( A ) µ ( B ) = µ ( A × B ) ( ⋆ ) = Z AB − µ ( A ∩ xB ) dµ = Z Z µ ( A ∩ xB ) dµ = α, as the equality ( ⋆ ) holds since | X × Y | = X x ∈ XY − | X ∩ xY | for any two finite subsets X and Y of a group. (cid:3) A standard translation using Łoś’s theorem (to avoid repetitions of such a trans-lation, see the proof of Proposition 4.10) yields the following finitary version:
Corollary 4.6. ( cf. [8, Lemma 5.1] & [2, Proposition 3]) For every positive δ , ǫ and η there is some integer d = d ( δ, ǫ, η ) such that for every finite d -quasirandomgroup G and subsets A and B of G of density at least δ , we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ G | | A ∩ xB | < (1 − η ) | A || B || G | (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) < ǫ | G | . The following result may be seen as a first attempt to solve complete amalga-mation problems, though restricting the conditions to those given by products.
Theorem 4.7.
Fix a natural number n ≥ . For each non-empty subset F of { , . . . , n } , let A F be a subset of the ultra-quasirandom group G of positive density.The set X n = { ( a , . . . , a n ) ∈ G n | a F ∈ A F for all ∅ 6 = F ⊆ { , . . . , n }} has measure Q F µ ( A F ) with respect to the normalized counting measure µ n on G n ,where a F stands for the product of all a i with i in F written with the indices inincreasing order.Proof. We reproduce Gower’s proof of [8, Theorem 5.3] and proceed by inductionon n . For n = 2 , set B = A and C = A , . A pair ( a, b ) satisfies all threeconditions if and only if a lies in A and b in B ∩ a − C . Thus µ ( X ) = Z A µ ( B ∩ a − C ) dµ Cor. . µ ( B ) µ ( C ) µ ( A ) , as desired. For the general case, for any a in A , set B F ( a ) = A F ∩ a − A ,F ,for ∅ 6 = F ⊂ { , . . . , n } . Corollary 4.5 yields that µ ( B F ( a )) = µ ( A F ) µ ( A ,F ) for µ -almost all a in A . A tuple ( a , . . . , a n ) in G n belongs to X n if and only if thetuple ( a , . . . , a n ) belongs to X n − ( a ) = (cid:8) ( x , . . . , x n ) ∈ G n − | x F ∈ B F ( a ) for all ∅ 6 = F ⊆ { , . . . , n } (cid:9) and a lies in A . By induction, the set X n − ( a ) has constant µ n − -measure Q F µ ( A F ) µ ( A ,F ) , where F now runs through all non-empty subsets of { , . . . , n } .Thus µ n ( X n ) = Z A µ n − ( X n − ( a )) dµ = µ ( A ) Y F µ ( A F ) µ ( A ,F ) = Y F µ ( A F ) , which yields the result. (cid:3) A standard translation using Łoś’s theorem (again, we refer to the proof of 4.10to avoid repetitions) yields the following finitary version, which was already presentin a quantitative form for n = 2 (setting A = A , B = A and C = A ) in Gowers’swork [8, Theorem 3.3]. Corollary 4.8. ( cf. [8, Theorem 5.3]) Fix a natural number n ≥ . For every ∅ 6 = F ⊆ { , . . . , n } let δ F > be given. For every η > there is some integer d = d ( n, δ F , η ) such that for every finite d -quasirandom group G and subsets A F of G of density at least δ F , we have that |X n | ≥ − η | G | n − − n Y F | A F | , where X n is defined as in Theorem 4.7 with respect to the group G . The above corollary yields in particular that |{ ( a, b, c ) ∈ A × B × C | ab = c }| > − η | G | | A || B || C | as first proved by Gowers [8, Theorem 3.3], which implies that the number of suchtriples is a proportion (uniformly on the densities and η ) of | G | . Digression: Local results on finite groups.
In the following aside, we willadapt some of the ideas present in the previous proof for quasirandom groups toarbitrary finite groups. The reader has certainly noticed that we have not used thefull strength of quasirandom groups in the proof of Corollary 4.8, but merely that G = G M in the proof of Corollary 4.5, which uses Corollary 3.5.Theorem 3.3 holds nevertheless in any definably amenable pair for any threeweakly random types which are product-compatible. Thus, it yields asymptotic in-formation for subsets of positive density in arbitrary finite groups satisfying certainregularity conditions, which force that in the ultraproduct any three completions arein a suitable position to apply Theorem 3.3. We will present two examples of suchregularity notions. Our intuition behind these notions is purely model-theoreticand we ignore whether they are meaningful from a combinatorial perspective.We now introduce the first notion, called regular position with respect to some ǫ > . Definition 4.9.
Three finite subsets A , B and C in a group G are in an ǫ -regular po-sition with respect to some ǫ > if the subsets A A − A , B B − B and C C − C are product-compatible , that is, (cid:0) ( A A − A ) · ( B B − B ) (cid:1) ∩ ( C C − C ) = ∅ for all subsets A of A , B of B and C of C of relative density at least ǫ .In the case of a d -quasirandom group, every triple of subsets of density at least δ is in regular position whenever ǫ ≥ / ( δ √ d ) [18, Proposition 0], so the condition inCorollary 4.4 is trivially fulfilled. On the other hand, in an arbitrary finite (abelian)group G , it is easy to find subsets of positive density which will not be in a suitableregular position, such as cosets of subgroups or more generally, hereditarily product-free subsets. The above notion provides a local version of quasirandomness, whichwill suffice to prove the following result. MALGAMATION AND ROTH’S THEOREM 17
Proposition 4.10.
For every δ > there are η > and ǫ > such that for everythree subsets A , B and C of a finite group G of density at least δ which are in ǫ -regular position, |{ ( a, b, c ) ∈ A × B × C | ab = c }| > η | G | . Proof.
The proof is a straight-forward application of Łoś’s theorem. Assuming thatthe statement does not hold, there is a fixed δ > such that for each m and n in N ,we find three subsets A m,n , B m,n and C m,n of a finite group G m,n , each of densityat least δ , which are in /n -regular position, yet the number of such triples is atmost | G m,n | /m .Following the approach of Example 1.3(a), we consider a suitable expansion L ofthe language of groups and regard each group G n,n as an L -structure M n . Choosea non-principal ultrafilter U on N and consider the ultraproduct M = Q U M n . Thelanguage L is chosen in such a way that the sets A = Q U A n,n , B = Q U B n,n and C = Q U C n,n are L -definable in the group G = Q U G n,n . Furthermore, thenormalised counting measure on G n,n induces a definable Keisler measure µ on G ,taking the standard part of the ultralimit. Choose now a countable elementarysubstructure M of M and note that the measure as well as the definable sets A , B and C are all definable over M . Note that the ultraproduct M is ℵ -saturated,so we can apply compactness over countable set of parameters, and thus over M . Claim.
Every two weakly random types over M containing the definable set A belong to the same coset of G M .Proof of Claim. Since the M -definable subsets A , B and C have positive density,fix two weakly random types q in B and r in C over M . Choose any weaklyrandom type p over M containing the definable set A . It suffices to show that C M ( p ) · C M ( q ) = C M ( r ) , so the coset C M ( p ) = C M ( r ) · C M ( q ) − will notdepend on the choice of p .Note that C M ( p ) ⊇ p · p − p , and likewise for q and r . If C M ( p ) · C M ( q ) = C M ( r ) , we obtain by compactness three M -definable subsets A of p , B of q and C of r such that (cid:0) ( A A − A ) · ( B B − B ) (cid:1) ∩ ( C C − C ) = ∅ . Without loss of generality, we may assume that A is a subset of A , and similarlyfor B and C with respect to B and C . By weakly randomness of the types p , q and r , there exists a common value n in N such that µ ( A ) ≥ µ ( A ) /n (and thesame for B and C ). Łoś’s theorem yields infinitely many natural numbers n ≥ n such that | A ( G n,n ) || A ( G n,n ) | , | B ( G n,n ) || B ( G n,n ) | , | C ( G n,n ) || C ( G n,n ) | ≥ n ≥ n , yet (cid:0) ( A A − A )( G n,n ) · ( B B − B )( G n,n ) (cid:1) ∩ ( C C − C )( G n,n ) = ∅ , which contradicts that the corresponding sets in G n,n were in /n -regular position. (cid:3) Claim
It follows from the above proof that all weakly random types of B over M liein the same coset, and similarly for C . Furthermore, any choice of weakly randomtypes p in A , q in B and r in C over M satisfies that C M ( p ) · C M ( q ) = C M ( r ) . Theorem 3.3 yields a realization a of p weakly random over M , b , where b realizes q , such that a · b realizes r . The pair ( a, b ) lies in U = { ( a, b ) ∈ A × B | a · b ∈ C } , which is in definable bijection over M with the corresponding collection of triples.Since a is weakly random over M , b and b is also weakly random over M , so is ( a, b ) weakly random over M by Lemma 1.6. This forces U to have positive densitywith respect the normalized pseudo-finite counting measure in G , which gives thedesired contradiction, since the limit with respect to U of the density of U ( G n,n ) is , by construction. (cid:3) As mentioned in the introduction of this aside, our motivation behind Definition4.9 of ǫ -regular position was to find a triple of weakly random types ( p, q, r ) in orderto apply Theorem 3.3, which yields in particular solutions of the equation x · y = z in p × q × r . Any weakly random type p in G M clearly gives raise to such a triple,namely the triple ( p, p, p ) . This naive observation is our main motivation behind thesecond notion of this excursus, which will impose that, in the ultraproduct, some(or rather, every) weakly random completion of our set of positive density will liein subgroup G M (or rather in G M for some countable elementary substructure M of the ultraproduct). Definition 4.11.
A finite subset A of a group G is product-rich up to ǫ if A · A ∩ A = ∅ whenever A ⊆ A has density at least ǫ in A .This notion will allow us to reproduce the proof of Corollary 4.8 in order toprovide a local version of [8, Theorem 5.3] to count the number of tuples suchthat all its possible products (enumerated in an increasing order) lie in a fixedproduct-rich set of positive density. Theorem 4.12.
Fix a natural number n ≥ and let δ F > for ∅ 6 = F ⊆ { , . . . , n } be given. There are ǫ = ǫ ( n, δ F ) > and η = η ( n, δ F ) > such that for every finitegroup G and subsets A F of G of density at least δ F which are all product-rich upto ǫ , we have that |{ ( a , . . . , a n ) ∈ G n | a F ∈ A F for all ∅ 6 = F ⊆ { , . . . , n }}| ≥ η | G | n , where a F stands for the product, enumerated in an increasing order, of all a i with i in F . In particular, setting A F = A for a fixed subset A of G of density at least δ whichis product-rich up to ǫ , we conclude that the proportion of the set of n -tuples in G n whose possible increasing products are all contained in A is positive (cf. [14,Theorem 3.7]). Proof.
The result follows immediately from the following claim by a standard ul-traproduct argument using Łoś’s theorem. We refer the reader to the proof ofProposition 4.10 for a guideline of the translation process from the infinite versionto the finitary statement.
Claim.
In a non-principal ultraproduct M of finite groups such that the normalisedpseudo-finite counting measure µ is definable over a countable submodel M , con-sider M -definable subsets A F , for ∅ 6 = F ⊆ { , . . . , n } of positive density which are MALGAMATION AND ROTH’S THEOREM 19 all M -definably (or internally) product-rich up to /m , for every m in N . There isa tuple ( a , . . . , a n ) in G n weakly random over M such that the product a F lies in A F for every subset F as above. Indeed, notice that an ultraproduct of product-rich subsets does not yield aproduct-rich subset A up to /m for all m in N , but only for M -definable (orinternal) subsets of A .The proof of the claim is by induction on n . The base case n = 2 and theinduction step have the exact same proof, so we assume that the statement of theClaim has already been shown for n − .Since the ultraproduct is ℵ -saturated, a straight-forward compactness argumentyields that all weakly random types over a countable elementary substructure M containing a fixed M -definable set A of positive density which is product-rich upto /m , for every m in N , must lie in G M . Thus, choose weakly random types p F in G M containing A F , for every subset F . Theorem 3.3 applied to each triple ofthe form ( p − F , p − , p − ,F ) yields for each ∅ 6 = F ⊆ { , . . . , n } a realization a of p such that B F = A F ∩ a − A ,F has positive density. We may assume that the realization a works simultaneouslyfor all subsets F , as the measure µ is definable over M and hence invariant.Notice that the set B F is not definable over M , yet Löwenheim-Skolem gives acountable submodel M containing M ∪ { a } . Since A F was product-rich up to /m , so is B F . By induction, we find a weakly random tuple ( a , . . . , a n ) over M such that the product a F lies in B F for every subset ∅ 6 = F ⊆ { , . . . , n } . Observethat for n = 2 , there is only one such subset, namely A ∩ a − A , . Lemma 1.6yields that the tuple ( a , . . . , a n ) is weakly random over M and by constructionthe product a F lies in A F for every subset ∅ 6 = F ⊆ { , . . . , n } , as desired. (cid:3) Solving equations and Roth’s theorem on progression
In this section, we will show how Theorem 3.3 yields immediately a proof ofRoth’s Theorem, by showing that a subset of positive density in a finite abeliangroup of odd order has a solution to the equation x + z = 2 y . In fact, as explainedin the introduction, our methods adapt to the non-abelian context and allow us tostudy more general equations such as x n · y m = z r for n + m = r . In particular,this yields the existence of non-trivial solutions of the equation x · z = y in finitegroups of odd order [1, Corollary 6.5] & [24, Theorem 1.2], though our methods arenon-quantitative.Now, we state and prove the following version of Theorem D from the introduc-tion. Theorem 5.1.
For every K ≥ and any natural numbers k, m ≥ there is some η = η ( K, k, m ) > with the following property: Given a subset A of small tripling K in an arbitrary group G and any three functions f , f and f from A to A ⊙ m ,each with fibers of size at most k , such that f ( a ) · f ( a ) = f ( a ) for all a ∈ A, then |{ ( a , a , a ) ∈ A × A × A | f ( a ) · f ( a ) = f ( a ) }| ≥ η | A | . To deduce Theorem C in the Introduction it suffices to set m = n n n and f i : x x n i for ≤ i ≤ . Proof.
As in the proof of Proposition 4.10, we proceed by contradiction using Łoś’stheorem. Assuming that the statement does not hold, there are K ≥ and k suchthat for each n in N , we find a subset A n of tripling K in a group G n , as well asfunctions f i,n : A n → A ⊙ mn , for ≤ i ≤ , of fibers at most k such that f ,n ( a ) · f ,n ( a ) = f ,n ( a ) for all a ∈ A n , yet the number of triples ( a , a , a ) in A n × A n × A n as above is at most | A n | /n .As before, a non-principal ultrafilter on N produces an ultraproduct M in asuitable language L which gives rise to a definable group G equipped with a dis-tinguished definable subset A such that ( G, A ) form a definably amenable pair asexplained in Example 1.3(b). Furthermore, we also obtain three definable functions f , f and f from A to A ⊙ m whose fibers have size at most k and such that f ( a ) · f ( a ) = f ( a ) for all a ∈ A. We now fix a countable elementary substructure M of the ultraproduct M andnote that the measure µ as well as the set A and the functions f i ’ s are all definableover M .Choose now a weakly random element a in A over M , and set p i = tp( f i ( a ) /M ) .Note that each type p i lies in h A i for ≤ i ≤ . Since a and f i ( a ) are in finite-to-onecorrespondence, the types p , p and p are again weakly random over M . Thefunctional equation of f , f and f implies that the cosets of h A i M of the p i ’s arecompatible: C M ( p ) · C M ( p ) = C M ( p ) . Theorem 3.3 yields a realizations b of p weakly random over M , b , with b realizing p , such that b · b belongs to f ( A ) . Write b i = f i ( a i ) for some a i in A ,and notice that a is weakly random over M , a (since f and f have finite fibers).As before, the pair ( a , a ) lies in the M -definable subset Λ = { ( x , x ) ∈ A × A | f ( x ) · f ( x ) ∈ f ( A ) } , which is in definable k -to- -correspondence over M with the collection of triples.Since a is weakly random over M , a , the set Λ has positive density in G × G with respect to the measure µ by Lemma 1.6, which gives the desired contradiction,since the ultralimit of the densities of Λ( G n ) is , by construction. (cid:3) Remark 5.2.
An inspection of the proof yields that the condition f ( a ) · f ( a ) = f ( a ) for all a ∈ A can be replaced by the condition that |{ a ∈ A | f ( a ) · f ( a ) = f ( a ) }| ≥ ǫ | A | for some constant ǫ > given beforehand, for this condition is sufficient to obtaina weakly random element a in A over M with f ( a ) · f ( a ) = f ( a ) .For sets of positive density, the functions f , f and f in the statement abovecan be taken from A to G . Therefore, a verbatim adaptation of the above proofyields the following: Corollary 5.3.
For every positive real numbers δ and ǫ , and every natural number k ≥ there is some η = η ( δ, ǫ, k ) > with the following property: Given any subset MALGAMATION AND ROTH’S THEOREM 21 A of a finite group G of positive density at least δ and three functions f , f and f from A to G , each with fibers of size at most k , and such that { a ∈ A | f ( a ) · f ( a ) = f ( a ) } has density at least ǫ , then |{ ( a , a , a ) ∈ A × A × A | f ( a ) · f ( a ) = f ( a ) }| ≥ η | G | . In particular, there is a non-trivial solution (that is, not of the form ( a, a, a ) ) in A to the equation f ( x ) · f ( x ) = f ( x ) , whenever G has size at least / ( ηδ ) .Observe that some compatibility condition on the equation is necessary for thestatements above to hold, as the equation x · y = z has no solution in a product-freesubset of density at least δ . Nonetheless, the strategy above permits to find solu-tions for this equation in some special circumstances, such as in ultra-quasirandomgroups. Another remarkable instance of solving equations in a group is Schur’sproof [26, Hilfssatz] on the existence of a monochromatic triangle in any finite col-oring (or cover) of the natural numbers , . . . , N , for N sufficiently large. In thisparticular case, the corresponding equation is again x · y = z . Sanders [25] re-marked that Schur’s original proof can be adapted in order to count the number ofmonochromatic triples ( x, y, x · y ) . Since any weakly random type p in G M mustdetermine a color and Theorem 3.3 applies to ( p, p, p ) , a standard application ofŁoś’s theorem along the lines of the proof of Proposition 4.10 yields the followingresult of Sanders. Theorem 5.4. ([25, Theorem 1.1])
For every natural number k ≥ there is some η = η ( k ) > with the following property: Given any coloring on a finite group G with k many colors A , . . . , A k , there exists some color A j , with ≤ j ≤ k , suchthat |{ ( a, b, c ) ∈ A j × A j × A j | a · b = c }| ≥ η | G | . Notice that the color A j as in the previous theorem will not be product-free, forthe equation x · y = z has a solution in A j . For ultra-quasirandom groups, no setof positive density is product-free. In fact Gowers showed a stronger version [8,Theorem 5.3] of Schur’s theorem, taking A F = A for ∅ 6 = F ⊆ { , . . . , n } with thenotation of Corollary 4.8.In Theorem 4.12, we provided a first local version of Gower’s result for arbi-trary (ultraproducts of) finite groups, under the additional assumption of product-richness. Our attempts to provide alternative proofs of Corollary 4.8 for arbitraryultraproducts of finite groups, without assuming ultra-quasirandomness, led us toisolate a particular instance of complete amalgamation problems (cf. the questionin the Introduction). Question.
Let M be a countable elementary substructure of a sufficiently saturateddefinably amenable pair ( G, X ) and p be a weakly random type in h X i M . Given anatural number n , is there a random tuple ( a , . . . , a n ) in G n such that a F realizes p for all ∅ 6 = F ⊆ { , . . . , n } , where a F stands for the product, enumerated in anincreasing order, of all a i with i in F ? At the moment of writing, we do not have a solid guess what the answer to theabove question will be. Nonetheless, if the question could be positively answered, it would imply, mimicking the proof of Theorem 4.12, a finitary version of Hindman’sTheorem [10].
Corollary 5.5.
If the above question has a positive answer, then for every naturalnumbers k and n there is some constant η = η ( k, n ) > such that in any coloringon a finite group G with k many colors A , . . . , A k , there exists some color A j , with ≤ j ≤ k such that |{ ( a , . . . , a n ) ∈ G n | a F ∈ A j for all ∅ 6 = F ⊆ { , . . . , n }}| ≥ η | G | n , where a F stands for the product, enumerated in an increasing order, of all a i with i in F . Though we cannot bound from below the number of monochromatic tuples ( a , . . . , a n ) as above, we can however show that they must exist, adapting theproof of Galvin and Glazer (see [3]) of Hindman’s Theorem [10]. Theorem 5.6.
For every natural numbers k and n there is a natural number m = m ( k, n ) such that in every finite group G of size at least m colored with k many colors A , . . . , A k , there exists some color A j , with ≤ j ≤ k , and a tuple ( a , . . . , a n ) in G n such that a F lies in A j for all ∅ 6 = F ⊆ { , . . . , n } , where a F stands for the product, enumerated in an increasing order, of all a i with i in F .Proof. By a standard ultraproduct argument using Łoś’s theorem, we need onlyshow that in every internal k -coloring A , . . . , A k of an ultraproduct G of finitegroups, we can find an infinite sequence all of whose finite products in an increasingorder are monochromatic of the same color A j .Let µ be the normalized counting measure of the group G , which we assume tobe definable without parameters as in Example 1.3. Since the ultraproduct G is ℵ -saturated, fix a countable elementary substructure M of G . Denote by ⊗ thecoheir product on the space of types S G ( M ) over a model M : Given two types p and q over M , let b realize q and a realize p such that tp( a/M, b ) is finitely satisfiableover M , and set p ⊗ q = tp( a · b/M ) . Claim.
There exist a weakly random type p over M which is idempotent with respectto the coheir product ⊗ , i.e. p ⊗ p = p .Proof of Claim. Note that the support S ( µ ) over M is a closed subset of the com-pact space of types over M , since by definition S ( µ ) = S G ( M ) \ [ { [ ψ ] | ψ ( x ) ∈ L M of density } . As the coheir product defined above is continuous in the first coordinate, namelythe map p p ⊗ q is continuous for each q ∈ S G ( M ) , we need only show that thesupport S ( µ ) is closed under coheir products in order to conclude by Ellis’s Lemma[4, Lemma 1] the existence of an idempotent weakly random type p over M , asdesired.Hence, given a and b two weakly random elements over M of G with tp( a/M, b ) finitely satisfiable over M , we want to show that a · b is weakly random over M .Otherwise, there is an M -definable subset X of density containing a · b , so wecan find an element m in M such that m · b lies in X . Equivalently, the element b lies in m − · X , which has again density , since the measure is invariant underleft translation. As m − · X is definable over M , the element b cannot be weaklyrandom, which yields the desired contradiction. (cid:3) Claim
MALGAMATION AND ROTH’S THEOREM 23
Fix now a weakly random type p over M which is an idempotent with respectto the coheir product. Since G is colored by the internal subsets A , . . . , A k , thetype p contains a color A j , for some ≤ j ≤ k . Thus, we need only show that forevery subset A in p , we find an infinite sequence ( a n ) n ∈ N in M all of whose finiteproducts lie in A ( M ) .The idempotence of p yields two realizations a and b of p , with tp( a/M, b ) finitelysatisfiable over M , such that a · b realizes p again. Hence both a and a · b lie in A , sowe find an element a in A ( M ) such that b lies in a − · A . Thus, the M -definable set B = A ∩ a − · A lies in p , so iterate the process and find now a in B ( M ) ⊆ A ( M ) such that B ∩ a − B lies in p . Recursively construct a sequence ( B n ) n ∈ N of M -definable subsets of A as well as a sequence ( a n ) n ∈ N such that B = A , the element a n is contained in B n ( M ) and B n +1 = B n ∩ a − n B n lies in the idempotent p .By induction on r , the finite product a i · · · a i r lies in B i if i < . . . < i r , so theset A contains all finite ordered products of the sequence ( a i ) i ∈ N , as desired. (cid:3) References [1] V. Bergelson, R. McCutcheon and Q. Zhang,
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