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COMPLETE TYPE AMALGAMATION AND ROTH’S THEOREM

ON 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 onproducts of three sets and yields model-theoretic proofs of existing asymptoticresults for quasirandom groups. In particular, we show the existence of non-quantitative lower bounds on the number of arithmetic progressions of length3 for subsets of small doubling in arbitrary abelian groups without involutions.

Introduction

Szemerédi answered positively a question of Erdős and Turán by showing [28]that every subset A of N with upper density

lim supn→∞

|A ∩ {1, . . . , n}|

n> 0

must contain an arithmetic progression of length k for every natural number k.For k = 3, the existence of arithmetic progressions of length 3 (in short 3-AP)was already proven by Roth in what is now called Roth’s theorem on arithmeticprogressions [21] (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[5], which however has upper density 0.

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, ab2) can be seen as a 3-AP, even for non-commutative groups. Gowers asked[6, Question 6.5] whether the proportion of pairs (a, b) in PSL2(q), for q a primepower, such that a, ab and ab2 all lie in a fixed subset A of density δ approximatelyequals δ3. Gower’s question was positively answered by Tao [30] and later extendedto arbitrary non-abelian finite simple groups by Peluse [19]. For arithmetic progres-sions (a, ab, ab2, ab3) of length 4 in PSL2(q), a partial result was obtained in [30],

Date: January 19, 2022.2010 Mathematics Subject Classification. 03C45, 11B30.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.

1

http://arxiv.org/abs/2009.08967v3

2 AMADOR MARTIN-PIZARRO AND DANIEL PALACÍN

whenever the element b is diagonalizable over the finite field Fq (which happenshalf of the time).

A different generalization of Roth’s theorem, present in work of Sanders [22]and Henriot [8], concerns the existence of a 3-AP in finite sets of small doublingin abelian 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 mostK, the comparable set A ∪ A−1 ∪ {idG} (of size at most 2|A| + 1) has triplingat most (CKC)2 with respect to some explicit absolute constant C > 0, 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 [10], 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 asimple theory, Hrushovski’s stabilizer of a generic type over an elementary sub-structure M is the connected component G00

M , that is, the smallest type-definablesubgroup over M of bounded index (bounded with respect to the saturation of theambient universal model). Generic types in G00

M are called principal types. If thetheory is stable, there is a unique principal type, but this need not be the case forsimple theories. However, Pillay, Scanlon and Wagner noticed [20, Proposition 2.2]that, given three principal types p, q and r in a simple theory over an elementarysubstructure M , there are independent realizations a of p and b of q over M suchthat ab realizes r. The main ingredient in their proof is a clever application of3-complete amalgamation (also known as the independence theorem) over the ele-mentary substructure M . For the purpose of the present work, we shall not definewhat a general complete amalgamation problem is, but a variation of it, restrictingthe problem to conditions given by products with respect to the underlying grouplaw:

Question. Fix a natural number n ≥ 2. For each non-empty subset F of {1, . . . , n},let pF be a principal generic (that is, weakly random) type over the elementarysubstructure M . Can we find (under suitable conditions) an independent (weaklyrandom) tuple (a1, . . . , an) of Gn such that for all ∅ 6= F ⊆ {1, . . . , n}, the elementaF realizes pF , where aF stands for the product of all ai, with i in F , written withthe indices in increasing order?

The above formulation resonates with [5, 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 1.4). As a by-product, we obtain the corresponding version of the resultof Pillay, Scanlon and Wagner (Corollary 3.4):

Main Theorem. 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

AMALGAMATION 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 equa-tions in subsets of small tripling, as well as to reprove model-theoretically someof the known results for ultra-quasirandom groups, that is, asymptotic limits ofquasirandom groups, already studied by Bergelson and Tao [2], and later by thesecond author [18]. In particular, in Corollary 4.8 we give a non-quantitative model-theoretic proof of Gower’s results [6, Theorem 3.3 & Theorem 5.3]. In Section 5,we further explore this analogy to extend some of the results of Gowers to a localsetting, without imposing that the group is an ultraproduct of quasirandom groups(see Theorem 5.7). Finally, in Section 6, setting q and r both equal to p in theMain Theorem, we can easily deduce a finitary (albeit non-quantitative) versionof Roth’s theorem on 3-AP for finite subsets of small doubling in abelian groupswith trivial 2-torsion (Corollary 6.3), which resembles previous work of Sanders [22,Theorem 7.1] and generalizes a result of Frankl, Graham and Rödl [3, Theorem 1].

Whilst almost all of the statements presented so far are of combinatorial nature,our proofs are model-theoretic. Hence, we will assume throughout the text a certainfamiliarity with basic notions in model theory. Sections 1, 2 and 3 contain themodel-theoretic core of the paper, whilst Sections 4, 5 and 6 contain applicationsto additive combinatorics.

Acknowledgements. We are most indebted to Julia Wolf for her patience andhelpful remarks which have considerably improved a previous version of this article.

1. Randomness and Fubini

Most of the material in this section can be found in [7, 10, 16, 25].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 Un is definable over the parameter set A if there exists a formulaφ(x1, . . . , xn, y1, . . . , ym) and a tuple a = (a1, . . . , am) in A such that an n-tuple bbelongs 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 subsetX 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 automorphismsAut(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 Lcontains the language of groups and the universe of our ambient model is a group.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 an underlying defin-able group G together with the following data:

• A definable subset X of G;


• The (boolean) ring R of definable sets contained in the subgroup 〈X〉 gen-erated by X , that is, the subcollection R is closed under finite unions andrelative set-theoretic differences;

• A finitely additive measure µ on R invariant under left and right translationwith µ(X) = 1.

Note that the subgroup 〈X〉 generated by the subset X need not be definable,but it is locally definable, for the subgroup 〈X〉 is a countable union of definablesets of the form

X⊙n = X1 · · ·X1︸︷︷︸

n

,

where X1 is the definable set X∪X−1∪{idG}. Furthermore, every definable subsetY of 〈X〉 is contained in some finite product X⊙n, by compactness and saturationof the ambient model.

Model-theoretic compactness implies that the finitely additive measure µ satisfiesCarathéodory’s criterion, so there exists a unique σ-additive measure on the σ-algebra generated by R. We will denote the extension again by µ, though therewill be (most likely) Borel sets of infinite measure, as noticed by Massicot andWagner:

Fact 1.2. ([16, Remark 4]) The subgroup 〈X〉 is definable if and only if µ(〈X〉) isfinite.

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 1.3. 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 ≥ 1 andevery ǫ > 0, there is a partition of the L-definable set

{y ∈ U|y| | ϕ(U, y) ⊆ X⊙n}

into L-formulae ρ1(y), . . . , ρm(y) such that whenever a pair (b, b′) in U|y| × U|y|

realizes ρi(y) ∧ ρi(y′), 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 subsetX⊙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). Notice that whenever the measure µ is definable, givena definable subset ϕ(x, b) of measure r and a value ǫ > 0, the tuple b lies in somedefinable subset which is contained in

{

y ∈ U|y| | r − ǫ ≤ µ(ϕ(U, y)) ≤ r + ǫ}

.

Assuming that µ is definable, its extension to the σ-algebra generated by the de-finable subsets of 〈X〉 is again invariant under left and right translations, as wellas under automorphisms: Indeed, every automorphism τ of Aut(U) (likewise fora translation) gives rise to a measure µτ , such that µτ (Y ) = µ(τ(Y )) for everymeasurable subset Y of 〈X〉. Since µτ agrees with µ on R, we conclude that theσ-additive measure µτ = µ by the uniqueness of the extension. Thus, the measure


of a Borel subset Y in the space of types containing a fixed definable set Z in Rdepends solely on the type of the parameters defining Y .

Example 1.4. Let (Gn)n∈N be an infinite family of groups, each with a distin-guished finite subset Xn. Expand the language of groups to a language L includinga unary predicate and set Mn to be an L-structure with universe Gn, equippedwith its group operation, and interpret the predicate as Xn. Following [10, Section2.6] we can further assume that L has predicates Qr,ϕ(y) for each r in Q≥0 andevery formula ϕ(x, y) in L such that Qr,ϕ(b) holds if and only if the set ϕ(Mn, b)is finite with |ϕ(Mn, b)| ≤ r|Xn|. Note that if the original language was countable,so is the extension L.

Consider now the ultraproduct M of the L-structures (Mn)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-formulaϕ(x, y) and every tuple b in U|y| such that ϕ(U, b) is a subset of 〈X〉, define

µ(ϕ(x, b)) = inf{r ∈ Q≥0 | Qr,ϕ(b) holds

},

where we assign ∞ if Qr,ϕ(b) holds for no value r. This is easily seen to be afinitely additive definable measure on the ring R of definable subsets of 〈X〉 which isinvariant under left and right translation. In particular, the pair (G,X) is definablyamenable.

We will throughout this paper consider two main examples:

(a) The set X equals G itself, which happens whenever the subset Xn = Gn forU-almost all n in N. The normalized counting measure µ defined above isa definable Keisler measure [13] on the pseudo-finite group G. Note that inthis case the ring of sets R coincides with the Boolean algebra of all definablesubsets of G.

(b) For U-almost all n, the set Xn has small tripling: there is a constant K > 0such that |XnXnXn| ≤ K|Xn| (or more generally |XnX

−1n Xn| ≤ K|Xn|).

The non-commutative Plünnecke-Ruzsa inequality [29, Lemma 3.4] yields that|X⊙m

n | ≤ KOm(1)|Xn|, so the measure µ(Y ) is finite for every definable subsetY of 〈X〉, since Y is then contained in X⊙m for some m in N. In particular,the corresponding σ-additive measure µ is again σ-finite.

Whilst each subset Xn in the example (b) must be finite, we do not impose thatthe groups Gn are finite. If the set Xn has tripling at most K, the set X⊙1 =Xn ∪ X−1

n ∪ {idG} has size at most 2|Xn| + 1 and tripling at most (CKC)2 withrespect to some explicit absolute constant C > 0. Thus, taking ultraproducts, bothstructures (G,X) and (G,X⊙1) 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 above example can be adapted to consider countable amenable groups.

Example 1.5. Recall that a countable group is amenable if it is equipped with asequence (Fn)n∈N of finite sets of increasing cardinalities (so lim

n→∞|Fn| = ∞) such

that all g in G,

limn→∞

|Fn ∩ g · Fn|

|Fn|= 1,


Such a sequence of finite sets is called a (left) Følner sequence. Notice that asubsequence of a Følner sequence is again Følner and so is the sequence (Fn×Fn)n∈N

in the group G×G.Given an amenable group G with a distinguished Følner sequence F = (Fn)n∈N

and a non-principal ultrafilter U on N, the ultralimit

µ(Y ) = limn→U

|Y ∩ Fn|

|Fn|,

induces a finitely additive measure on the Boolean algebra of subsets of G which isinvariant under left and right translation. Starting from a fixed countable languageL expanding the language of groups, we can render the above measure definable,similarly as in Example 1.4. Hence, we can consider G as a definably amenablepair, setting X = G.

Example 1.6. Every stable group G is fsg and thus equipped with a unique leftand right translation invariant Keisler measure which is generically stable (see [12]& [25, Example 8.34.]).

Similarly, a compact semialgebraic Lie groupG(R), or more generally a definablycompact group G definable in an o-minimal expansion of a real closed field is againfsg. If the group is the R-rational points of a compact semialgebraic Lie group, thismeasure coincides with the normalised Haar measure.

Hence, we can consider in these two previous cases (stable and o-minimal com-pact) the group G as a definably amenable pair, setting X = G.

If a group G is definable, so is every finite cartesian product. Moreover, theconstruction in Example 1.4 and 1.5 can also be carried out for a finite cartesianproduct to produce for every n ≥ 1 in N a definably amenable pair (Gn, Xn), where〈Xn〉 = 〈X〉n, equipped with a definable σ-finite measure µn. Thus, the followingassumption is satisfied by our examples 1.4, 1.5 and 1.6.

Assumption 1. For every n ≥ 1, the pair (Gn, Xn) is definably amenable for adefinable σ-finite measure µn in a compatible fashion: the measure µn+m extendsthe corresponding product measure µn × µm.

The definability condition in Definition 1.3 implies that the function

Sm(C) → R

tp(b/C) 7→ µn(ϕ(x, b))

is well-defined and continuous for every LC -formula ϕ(x, y) with |x| = n and |y| = msuch that ϕ(x, y) defines a subset of 〈X〉n+m. Therefore, for such LC -formulaeϕ(x, y), we can consider the following measure ν on 〈X〉n+m,

ν(ϕ(x, y)) =

∫

〈X〉mµn(ϕ(x, y)) dµm,

where the integral runs in fact over the LC -definable subset {y ∈ 〈X〉m | ∃xϕ(x, y)}.For the pseudo-finite measures described in Example 1.4, the above integral equalsthe ultralimit

limk→U

1

|Xk|m

∑

y∈〈Xk〉m

|ϕ(x, y)|

|Xk|n,


so ν equals µn+m and consequently Fubini-Tonelli holds, see [2, Theorem 19]. Thesame holds whenever the measure is given by densities with respect to a Følner se-quence in an amenable group, as in Example 1.5. For arbitrary definably amenablepairs, whilst the measure ν extends the product measure µn × µm, it need not bea priori µn+m [27, Remark 3.28]. Keisler [13, Theorem 6.15] exhibited a Fubini-Tonelli type theorem for general Keisler measures under certain conditions. Theseconditions hold for the unique generically stable translation invariant measure of anfsg group, see Example 1.6. We will impose a further restriction on the definablyamenable pairs we will consider, taking Examples 1.4, 1.5 and 1.6 as a guideline.

Assumption 2. For every definably amenable pair (G,X) and its correspondingcompatible system of definable measures (µn)n∈N on the Cartesian powers of 〈X〉,the Fubini condition holds: Whenever a definable subset of 〈X〉n+m is given by anLC -formula ϕ(x, y) with |x| = n and |y| = m, the following equality holds:

µn+m(ϕ(x, y)) =

∫

〈X〉mµn(ϕ(x, y)) dµm =

∫

〈X〉nµm(ϕ(x, y)) dµn.

(Note that the above integrals do not run over the locally definable sets 〈X〉m and〈X〉n, but rather over definable subsets, since ϕ(x, y) is itself definable).

Whilst this assumption is stated for definable sets, it extends to certain Borelsets, whenever the language LC is countable.

Remark 1.7. Assume that LC is countable and fix a natural number k ≥ 1.For every Borel set Z(x, y) with |x| = n and |y| = m contained as a subset in(X⊙k)n+m, we have that the function y 7→ µ(Z(x, y)) is Borel, thus measurable,by the definability of the measure as well as the monotone convergence Theorem.Furthermore, the following identity holds:

µn+m(Z(x, y)) =

∫

〈X〉mµn(Z(x, y)) dµm =

∫

〈X〉nµm(Z(x, y)) dµn,

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.

Remark 1.8. The examples listed in Examples 1.4, 1.5 and 1.6 satisfy both As-sumptions 1 and 2.

Henceforth, the language is countable and all definably amenable pairs

satisfy Assumptions 1 and 2.

Adopting the terminology from additive combinatorics, we shall use the worddensity for the value of the measure of a subset in R of a definably amenable pair(G,X).

A (partial) type is said to be weakly random if it contains a definable subset in Rof positive density but no definable subset in R of density 0. Note that every weaklyrandom partial type Σ(x) over a parameter set A implies the definable set X⊙k inR for some k in N and thus it can be completed to a weakly random complete typeover any arbitrary set B containing A, since the collection of formulae

Σ(x) ∪{X⊙k \ Z |Z in R is B-definable of density 0

}


is finitely consistent. Thus, weakly random types exist (yet the partial type x = xis not weakly random whenever G 6= 〈X〉). As usual, we say that an element b ofG is weakly random over A if tp(b/A) is.

Weakly random elements satisfy a weak notion of transitivity.

Lemma 1.9. 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 〈X〉n+m containing thepair (a, b) has positive density with respect to the product measure µn+m, wheren = |a| and m = |b|. Since a is weakly random over C, b, the fiber Zb of Z over b hasmeasure µn(Zb) = 2r for some real number 0 < r. Hence b belongs a C-definablesubset Y of

{y ∈ Um | r ≤ µn(Zy) ≤ 3r} ,

by the definability of the measure. In particular, the measure µm(Y ) > 0. Thus,

µn+m(Z) =

∫

〈X〉mµn(Zy) dµm ≥

∫

Y

µn(Zy) dµm ≥ µm(Y )r > 0,

as desired. �

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. Though Random types alreadyappear in [11, Exercise 2.25], let us recall Hrushovski’s definition of ω-randomness.

Definition 1.10. Let Y in R be definable over the countable subset of parametersC (so Y ⊆ (X⊙k) for some k in N). We define inductively on n in N the Boolean

algebra DefYn (C) of subsets of Y of higher measurable complexity. The collection

DefY0 (C) consists of the LC -definable subsets of cartesian powers of Y , whereas

DefYn+1(C) is the Boolean algebra generated by both DefYn (C) and all the sets ofthe form

{a ∈ Y m | µℓ(Za) = 0},

where Z ⊆ Y ℓ+m runs over all subsets of DefYn (C).

Note that the every subset in DefYn (C) is Borel, so we can talk about their valuewith respect to the extensions of our original collection of σ-finite measures µk.However, the algebra DefY1 (C) contains new sets which are neither type-definablenor their complement is.

Definition 1.11. Let Y in R be definable over the countable subset of parametersC. A tuple of elements in Y is random over C if it lies in no subset Z of DefYn (C)of measure 0, 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 elements ofY is random over C, then the tuple a is in particular weakly random over C, whichjustifies our choice of terminology (instead of using the term wide type).

Notice that all the Boolean algebras DefYn (C) are countable. Hence, since the

value of the measure and its extension coincide for subsets of DefY0 (C), it follows

by σ-additivity of the measure that no subset of DefY0 (C) of positive measure canbe covered by Borel subsets of measure 0 from higher Defn’s, allowing to concludethe following result:


Remark 1.12. Every definable subset of 〈X〉 over the countable set C of positivedensity contains a random element over C.

Randomness is a symmetric notion.

Lemma 1.13. ([11, Exercise 2.25]) Let Y in R be definable over the countablesubset of parameters C. A finite tuple (a, b) of elements in Y is random over C ifand only if b is random over C and a is random over C, b.

Proof. Assume that (a, b) is random over C. Clearly so is b, thus we need onlyprove that a is random over C, b. Suppose on the contrary that there is a subsetZb of DefYn (C, b), for some n in N, of density 0 containing a. Write Z(x, b) = Zbfor some subset Z of Y |a|+|b| in DefYn (C). Thus, the pair (a, b) belongs to

Z̃ = Z ∩ {(x, y) ∈ Y |a|+|b| | Zy has density 0},

which is a subset in DefYn+1(C), and thus it cannot have density 0. However,Remark 1.7 yields

0 < µ|a|+|b|(Z̃) =

∫

Y |b|

µ|a|(Z̃b) dµ|b| = 0,

which gives the desired contradiction.Assume now that b is random over C and a is random over C, b. A verbatim

translation, using Remark 1.7, yields that whenever (a, b) lies in a finite density

subset Z of DefYn (C), then Z has positive measure. �

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 as well as to definably compact groups definable in o-minimal expan-sions of real closed fields.

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 (ai)i∈N overC with a0 = a such that the intersection⋂

i ϕ(x, ai) is empty. Archetypal examples of dividing formulae are of the form x = afor 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 fomulaψ(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 typedivides, 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 〈X〉 of positive density can divide, thus no weakly randomtype forks 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 A-invariant relationR(x, y) is stable if there is noA-indiscerniblesequence (ai, bi)i∈N such that

R(ai, bj) holds if and only if i < j.

A straight-forward Ramsey argument yields that the collection of invariant sta-ble relations is closed under Boolean combinations. Furthermore, an A-invariantrelation is stable if there is no A-indiscernible sequence as in the definition of lengthsome fixed infinite ordinal.

The following remark will be very useful in the following sections.

Remark 2.3. ([10, 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′/Mb) does not fork.(c) The relation R(a′, b) holds, whenever a′ ≡M a and tp(b/Ma′) 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. ([10, Lemma 2.10 & Proposition 2.25]) Given a real number αand LM -formulae ϕ(x, z) and ψ(y, z) with parameters over an elementary substruc-ture M , the M -invariant relation on the definably amenable pair (G,X)

Rαϕ,ψ(a, b) ⇔ µ|z|

(ϕ(a, z) ∧ ψ(b, z)

)= α

is stable. In particular, for any partial types Φ(x, z) and Ψ(y, z) over M , the relation

QΦ,Ψ(a, b) ⇔ Φ(a, z) ∧Ψ(b, z) is weakly random

is stable.

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 some 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 stabilizer

theorem 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 〈X〉00M the intersection of all subgroupsof 〈X〉 type-definable over M and of bounded index.

If a subgroup of bounded index type-definable overM exists, the subgroup 〈X〉00Mis again type-definable over M and has bounded index, see [10, Lemmata 3.2 & 3.3].Furthermore, it is also normal in 〈X〉 (cf. [10, Lemma 3.4]), since it is the kernel ofthe group homomorphism

〈X〉 → Sym(〈X〉/〈X〉00M )

g 7→ σg

where σg is the permutation mapping h〈X〉00M → gh〈X〉00M .


Fact 2.6. ([10, Theorem 3.5] & [17, Theorem 2.12]) Let (G,X) be a definablyamenable pair and let M be an elementary substructure. The subgroup 〈X〉00Mexists and equals

〈X〉00M = (p · p−1)2,

for any weakly random type p over M , where we identify a type with its realizationsin the ambient structure U. Furthermore, the set pp−1p is a coset of 〈X〉00M . Forevery element a in 〈X〉00M weakly random over M , the partial type p∩a ·p is weaklyrandom.

If the definably amenable pair we consider happens to be as in the first case ofExample 1.4 or a stable group as in Example 1.6, our notation coincides with theclassical notation G00

M .Note that each coset of 〈X〉00M is type-definable over M and hence M -invariant,

though it need not have a representative inM . Thus, every type p overM containedin 〈X〉 must determine a coset of 〈X〉00M . We denote by CM (p) the coset of 〈X〉00Mof 〈X〉 containing some (and hence every) realization of p.

3. On 3-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 countable elementary substructure M . We denote bySM (µ) the support of µ, that is, the collection of all weakly random types over Mcontained in 〈X〉.

Lemma 3.1. Given M -definable subsets A and B of 〈X〉 of positive density, thereexists some random element g over M with µ(Ag ∩B) > 0.

Proof. By Remark 1.12, let c be random in B over M and choose now g−1 inc−1A 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. �

Remark 3.2. Notice that the above results yields the existence of an element hrandom over M such that hA∩B, and thus A∩h−1B, has positive density: Indeed,apply the statement to the definable subsets B−1 and A−1.

The next result was first observed for principal generic types in a simple theoryin [20, 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 [18, 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 SM (µ) over thecountable elementary substructure M , there are realizations a of p and b of q witha weakly random over M, b and a · b realizing r if and only if their cosets over Msatisfy that CM (p) · CM (q) = CM (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 CM (p)·CM (q) = CM (r).We proceed by proving the following auxiliary claims.


Claim 1. Given finitely many M -definable subsets A1, . . . , An in p and B1, . . . , Bnin r, there exists a random element g in 〈X〉 over M with Aig ∩ Bj of positivedensity for all 1 ≤ i, j ≤ n.

Proof of Claim 1. The M -definable subsets A =⋂

1≤i≤nAi and B =⋂

1≤i≤n Bi liein p and r respectively, so they both have positive density. Lemma 3.1 applied toA and B yields the desired random element g. �Claim 1

Claim 2. There exists some element g in 〈X〉 weakly random over M such thatthe partial type p · g ∩ r is weakly random.

Proof of Claim 2. Set Y = X⊙2m for some natural number m such that the sym-metric set X⊙m contains all realizations of p and r. Consider now the clopen subset[Y ] in the space of types over M and note that the collection of weakly randomtypes over M containing the definable subset Y is a closed subset [Y ] ∩ SM (µ) of[Y ].

Since both the language and the elementary substructure M are countable, enu-merate the definable subsets (An)n∈N of the type p and (Bn)n∈N of the type r ina decreasing way, that is, with An+1 ⊆ An and Bn+1 ⊆ Bn for every n. By Claim1, there is for every n in N a type tp(gn/M) in [Y ] ∩ SM (µ) such that Angn ∩ Bnhas positive measure, say larger than δn for some rational number 0 < δn ≤ 1. Bycontinuity of the measure, the element gn lies in the M -type-definable subset

Zn = {y ∈ Y | µ(Any ∩Bn) ≥ δn}.

The corresponding closed set [Zn] contains the type tp(gn/M), which is an elementof [Y ] ∩ SM (µ). Thus, by compactness, the decreasing intersection

[Y ] ∩ SM (µ) ∩⋂

n∈N

[Zn]

is non-empty, so there exists a type tp(g/M) in the above intersection. By con-struction, the element g is weakly random over M . Also, given any two naturalnumbers n and m, we may assume that n ≤ m, so

µ(Ang ∩Bm) ≥ µ(Amg ∩Bm) ≥ δm > 0,

which yields that p · g ∩ r is weakly random, as desired. �Claim 2

Since CM (r) = CM (p) · CM (q), observe that any element g as in Claim 2 lies inCM (q). Fix now such a weakly random element g over M and choose a realizationb of q weakly random over M, g. Since weakly random types do not fork, note thattp(bg−1/M, g) does not fork over M .

Claim 3. For some g1 in 〈X〉 weakly random over M, g, b, the type p · (bg−1g1)∩ ris weakly random. In particular the type tp(g1/M, b, g) does not fork over M .

Proof of Claim 3. Since the weakly random type s = tp(g/M) lies in CM (q), thedifference bg−1 is a weakly random element in the normal subgroup 〈X〉00M . Hence,the partial type s ∩ bg−1s is weakly random over M, bg−1 by Fact 2.6. Choose anelement g1 realizing s weakly random over M, g, b such that bg−1g1 ≡M g as well.By invariance of the measure, we have that p · (bg−1g1) ∩ r is weakly random, asdesired. �Claim 3


Summarizing, the relation

Qp,r(u, v) ⇔ “p · (u · v) ∩ r is weakly random”

holds for the pair (bg−1, g1) with tp(g1/M, bg−1) non-forking over M . Note that theabove relation is stable, by Proposition 2.4, so Qp,r must hold for any pair (w, z)such that

w ≡M bg−1 , z ≡M g1 and tp(w/M, z) non-forking over M,

by the Remark 2.3. Setting w = bg−1 and z = g, we conclude that

p · b ∩ r = p · (bg−1g) ∩ r

is weakly random over M . Choose now a realization c of p · b ∩ r weakly randomover M, b and set a = cb−1, which realizes a weakly random extension of p to M, bby our choice of c. �

Corollary 3.4. Given three weakly random types p, q and r in 〈X〉00M , the partialtype

{(x, y) ∈ p× q |xy ∈ r}

is weakly random in the definably amenable pair (G2, X2).

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 1.9. �

It follows from Lemma 3.1 that there exists an element g in 〈X〉 for any twodefinable subsets A and B of positive density such that the intersection A∩gB hasagain positive density. We will now see that this density is constant within a cosetof 〈X〉00M .

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 gand h over M within the same coset of 〈X〉00M .

Proof. Without loss of generality, it suffices to consider the case where the valueµ(A ∩ gB) = α > 0 for some weakly random element g over M and denote byr its type over M . Choose some weakly random type p in 〈X〉00M over M . Byconstruction

CM (r) = CM (p) · CM (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 thatα = µ(c−1A ∩ dB).

For any weakly random type s = tp(h/M) in CM (r), we clearly have thatCM (s) = CM (r), so Theorem 3.3 yields that h = c1d1 for some realizations c1of p and d1 of r with tp(c1/M, d1) weakly random (thus non-forking over M). Asthe relation

RαA,B(u, v) ⇔ “µ(u−1A ∩ vB) = α”

is stable by Proposition 2.4, we conclude by the Remark 2.3 that µ(A ∩ hB) = α ,as desired. �


4. Ultra-quasirandomness revisited

Given a definably amenable pair (G,X) with 〈X〉 = G, a straight-forward ap-plication of compactness yields that X⊙n = G for some natural number n in N, soX generates G in finitely many steps. Up to scaling the σ-finite measure, we mayassume that G = X , so µ(G) = 1. This observation, together with Examples 1.4(a)and 1.6, motivates the following notion.

Definition 4.1. Let (G,X) be a definably amenable pair with X = G. We saythat the pair is generically principal if G = G00

M for some elementary substructureM .

In an abuse of notation, we will simply say that the group G is generically principal.

Remark 4.2. By [14, Corollary 2.6], a group G is generically principal if and onlyif G = G00

M for every elementary substructure M , so we may assume that M iscountable.

Example 4.3. Three known classes of groups are generically principal:

• Connected stable groups, such as every connected algebraic group over analgebraically closed field.

• Simple definably compact groups definable in some o-minimal expansion ofa real closed field, such as SOn(R) or PSLn(R).

• Ultra-quasirandom groups, introduced by Bergelson and Tao [2]. Let usbriefly recall this notion. A finite group is d-quasirandom, with d ≥ 1,if all its non-trivial representations have degree at least d. An ultra-product of finite groups (Gn)n∈N with respect to a non-principal ultra-filter U is said to be ultra-quasirandom if for every integer d ≥ 1, the set{n ∈ N |Gn is d-quasirandom} belongs to U .

The work of Gowers [6, Theorem 3.3] yields that every definable subsetA of positive density of an ultra-quasirandom group G(M) is not product-free, i.e. it contains a solution to the equation xy = z, and thus the sameholds in every elementary extension. Therefore, definability of the measureµ yields that G = G00

N over any elementary substructure N [14, Corollary2.6], so ultra-quasirandom groups are generically principal.

Theorem 3.3 and its corollaries yield now a short proof of the result mentionedin the above paragraph.

Lemma 4.4. The following conditions are equivalent for a definably amenable pair(G,G):

(a) The group G is generically principal.(b) For any three definable subsets A, B and C of positive density, we have that

G = A ·B · C and the set G \A · B−1 has measure 0.(c) There is no definable product-free set of positive density.

Proof. For (a) ⇒ (b): Given three subsets A, B and C of positive density definableover some countable elementary substructure M0, we need only show that everyelement g in G(M0) lies in A ·B ·C, which follows immediately from Corollary 3.4by choosing weakly random types p in A, q in B and r in gC−1 over M0.

If the M0-definable subset G\AB−1 had positive density, we could find a weaklyrandom type r over M0 containing this set. Any choice of weakly random types p


in A and q in B over M0 gives the desired contradiction by Theorem 3.3, since G00M0

equals G.The implication (b) ⇒ (c) is clear, taking A and B−1 to be the same set. Thus,

we are left to consider the implication (c) ⇒ (a). Suppose that G 6= G00M0

forsome countable elementary substructure M0 and take a weakly random type p ina non-trivial coset CM0

(p) of G00M0

. Note that p−1 · p · p ⊆ CM0(p). A standard

compactness argument yields the existence of some definable set A in p such thatidG does not lie in A−1 ·A ·A, so A is product-free. Since p is weakly random, thedefinable subset A has positive density. �

The following result on weak mixing, already present as is in the work of Taoand Bergelson, was implicit in the work of Gowers [6]. It will play a crucial role tostudy some instances of complete amalgamation of equations in a group.

Corollary 4.5. (cf. [2, Lemma 33]) Let G be a generically principal group. Giventwo definable subsets A and B of positive density,

µ(A ∩ gB) = µ(A)µ(B)

for µ-almost all elements g.

Proof. As before, fix some countable elementary substructure M0 such that both Aand B are M0-definable. We may assume that the measure µ is also definable overM0. By Corollary 3.5, let α be the value of µ(A ∩ gB) for some (or equivalently,every) weakly random element g over M0. Notice that α > 0 by the Remark 3.2.

In particular, the subset

Z = {x ∈ AB−1 | µ(A ∩ xB) = α}

is type-definable overM0 and contains all weakly random elements overM0. Clearly,the measure µ(Z) ≤ µ(AB−1) and the latter equals 1, by Lemma 4.4. If µ(Z) <

µ(AB−1), there is a M0-definable set Z̃ with Z ⊆ Z̃ ⊆ AB−1 such that µ(AB−1 \Z̃) > 0. Thus, the set AB−1 \ Z̃ has positive density and it must contain aweakly random element over M0, which gives the desired contradiction, so µ(Z) =µ(AB−1) = 1.

If we now denote by µ2 the normalized measure in G2, an easy computationyields that

µ(A)µ(B) = µ2(A × B) =

∫

AB−1

µ(A ∩ xB) dµ =

∫

Z

µ(A ∩ xB) dµ = α,

as desired. �

A standard translation using Łoś’s theorem yields the following finitary version:

Corollary 4.6. (cf. [6, Lemma 5.1] & [2, Proposition 3]) For every positive δ, ǫand 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

∣∣{x ∈ G

∣∣ |A ∩ xB||G| < (1− η)|A||B|

}∣∣ < ǫ|G|.

Proof. Assume for a contradiction that the statement does not hold, so there aresome fixed positive numbers δ, ǫ and η such that for each natural number d we findtwo subsets Ad Bd of a finite d-quasirandom group Gd, each of density at least δ,such that the cardinality of the subset

X (Gd) = {x ∈ Gd | |Ad ∩ xBd||Gd| < (1 − η)|Ad||Bd|}


is at least ǫ|Gd|.Following the approach of Example 1.4(a), we consider a suitable expansion L

of the language of groups and regard each group Gd as an L-structure Md. Choosea non-principal ultrafilter U on N and consider the ultraproduct M =

∏

U Md. Thelanguage L is chosen in such a way that the sets A =

∏

U Ad and B =∏

U Bdare L-definable in the ultra-quasirandom group G =

∏

U Gd. Furthermore, thenormalised counting measure on Gd induces a definable Keisler measure µ on G,taking the standard part of the ultralimit. By Corollary 4.5, for µ-almost all g inG, we have µ(A ∩ gB) = µ(A)µ(B). Hence the definable set

{x ∈ G

∣∣ (1− η)µ(A)µ(B) ≤ µ(A ∩ xB) ≤ (1 + η)µ(A)µ(B)

}

has measure 1. Thus its complement, which is again definable, has measure 0,so in particular it has measure less than the fixed value ǫ. By Łoś’s theorem,it follows that |X (Gd)| < ǫ|Gd| for infinitely many d’s, which yields the desiredcontradiction. �

The following result is a verbatim adaption of [6, Theorem 5.3] and may be seenas a first attempt to solve complete amalgamation problems, though restricting theconditions to those given by products.

Theorem 4.7. Fix a natural number n ≥ 2. For each non-empty subset F of{1, . . . , n}, let AF be a subset of positive density of the generically principal groupG. The set

Xn = {(a1, . . . , an) ∈ Gn | aF ∈ AF for all ∅ 6= F ⊆ {1, . . . , n}}

has measure∏

F µ(AF ) with respect to the measure µn on Gn, where aF stands forthe product of all ai with i in F written with the indices in increasing order.

Proof. We reproduce Gower’s proof of [6, Theorem 5.3] and proceed by inductionon n. For n = 2, set B = A{2} and C = A{1,2}. A pair (a, b) lies in X2 if and only

if a belongs to A{1} and b to B ∩ a−1C. Thus

µ2(X2) =

∫

A{1}

µ(B ∩ a−1C) dµCor. 4.5

= µ(B)µ(C)µ(A{1}),

as desired. For the general case, for any a in A{1}, set BF1(a) = AF1

∩ a−1A1,F1,

for ∅ 6= F1 ⊂ {2, . . . , n}. Corollary 4.5 yields that µ(BF1(a)) = µ(AF1

)µ(A1,F1) for

µ-almost all a in A{1}. A tuple (a1, . . . , an) in Gn belongs to Xn if and only if thefirst coordinate a1 lies in A{1} and the tuple (a2, . . . , an) belongs to

Xn−1(a1) ={(x2, . . . , xn) ∈ Gn−1 | xF1

∈ BF1(a1) for all ∅ 6= F1 ⊆ {2, . . . , n}

}.

By induction, the set Xn−1(a) has constant µn−1-measure∏

F1µ(AF1

)µ(A1,F1),

where F1 now runs through all non-empty subsets of {2, . . . , n}. Thus

µn(Xn) =

∫

A1

µn−1(Xn−1(a1)) dµ = µ(A1)∏

F1

µ(AF1)µ(A1,F1

) =∏

F

µ(AF ),

which yields the desired result. �

A standard translation using Łoś’s theorem (we refer to the proof of Corollary4.6 to avoid repetitions) yields the following finitary version, which was alreadypresent in a quantitative form for n = 2 (setting A = A1, B = A2 and C = A12) inGowers’s work [6, Theorem 3.3].


Corollary 4.8. (cf. [6, Theorem 5.3]) Fix a natural number n ≥ 2. For every∅ 6= F ⊆ {1, . . . , n} let δF > 0 be given. For every η > 0 there is some integerd = d(n, δF , η) such that for every finite d-quasirandom group G and subsets AF ofG of density at least δF , we have that

|Xn| ≥1− η

|G|2n−1−n

∏

F

|AF |,

where Xn 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}| >1− η

|G||A||B||C|

as first proved by Gowers [6, Theorem 3.3], which implies that the number of suchtriples is a proportion (uniformly on the densities and η) of |G|2.

To conclude this section we answer affirmatively the question in the introductionfor generically principal groups, whenever all the types are based over a commoncountable elementary substructure.

Theorem 4.9. Fix a natural number n ≥ 2 and a countable elementary substruc-ture M0 of the generically principal group G. For each non-empty subset F of{1, . . . , n}, let pF be a weakly random type over M0. There exists a weakly randomn-tuple (a1, . . . , an) in Gn such that aF realises pF for all ∅ 6= F ⊆ {1, . . . , n},where aF stands for the product of all ai with i in F written with the indices inincreasing order.

Proof. Since M0 is countable, enumerate all the formulae occurring in each type pFin a decreasing way, that is, write pF = {AF,k}k∈N with AF,k+1 ⊆ AF,k for everynatural number k. We want to show that the set

Xn = {(x1, . . . , xn) ∈ Gn | pF (xF ) for all ∅ 6= F ⊆ {1, . . . , n}}

is weakly random, that is, we need to prove that the partial type

{¬ψ(x1, . . . , xn)}ψ∈Σ ∪ {xF ∈ AF,k}F∈Pk∈N

is consistent, where P = P({1, . . . , n})\{∅} and Σ is the set of LM0-formulae of µn-

measure 0. By compactness, since the subsets AF,k are enumerated decreasingly,we need only consider a finite subset of the above partial type where the level k0 isthe same for each of the subsets AF,k0 of positive density. By Theorem 4.7 the set

Xn,k0 = {(a1, . . . , an) ∈ Gn | aF ∈ AF,k0 for all ∅ 6= F ⊆ {1, . . . , n}}

has µn-measure∏

F µ(AF,k0) > 0, so we conclude the desired result. �

5. Local ultra-quasirandomness

In this section, we will adapt some of the ideas present in the previous sectionto arbitrary finite groups.

Theorem 3.3 holds in any definably amenable pair for any three weakly ran-dom types, whenever the types (or rather their cosets modulo G00

M ) are product-compatible. Thus, it yields asymptotic information for subsets of positive densityin arbitrary finite groups satisfying certain regularity conditions, which force thatin the ultraproduct any three completions are in a suitable position to apply ourmain Theorem 3.3. We will present a local example of such a regularity notion.


Our intuition behind this notion is purely model-theoretic and we ignore whetherit is meaningful from a combinatorial perspective.

In order to find particular solutions of the equation x · y = z in p× q × r usingTheorem 3.3, we start with a naive observation: a weakly random type p in G00

M

clearly gives raise to a suitable triple of types, namely the triple (p, p, p). This isour main motivation behind the following definition, which will impose that in theultraproduct, some weakly random completion of our set of positive density will liein subgroup G00

M (or rather in G00M0

for some countable elementary substructure M0

of the ultraproduct). We would like to express our gratitude to Julia Wolf (andindirectly to Tom Sanders) for pointing out that our previous definition of principalsubsets did not extend to the abelian case.

Definition 5.1. Fix ǫ > 0 and k in N. A finite subset A of a group G is (k, ǫ)-principal if

|A ∩ (Y · Y )| ≥ ǫ|A|

whenever Y is a neighborhood of the identity (that is, the set Y is symmetricand contains the identity) such that k many left translates (or equivalently, righttranslates) of Y cover A · A−1 ·A ·A−1.We shall say that the finite subset A is hereditarily (k, ǫ)-principal up to ρ if all itssubset of relative density at least ρ (in A) are (k, ǫ)-principal.

Example 5.2. Consider the finite group G = Zn × Z2. The set G is clearly(k, 1/k)-principal for every natural number k 6= 0, yet it is not hereditarily (2, 1/k)-principal up to 1/2 for any k 6= 0, for the subset A = Zn × {1̄} does not intersectY = Zn × {0̄}, which covers G in 2 steps.

Example 5.3. Given a subset A of a finite group G of density at least ǫ, thesymmetric set AA−1 is (k, ǫ/k)-principal. Indeed, if Y is a given neighborhood ofthe identity such that k many right translates of Y cover (AA−1)4, then there existssome c in G such that |Ac ∩ Y | ≥ |A|/k and so |AA−1 ∩ Y Y | ≥ ǫ|AA−1|/k, since(Ac ∩ Y )(Ac ∩ Y )−1 ⊆ AA−1 ∩ Y Y .

The above definition extends naturally to definably amenable pairs as follows:

Definition 5.4. Let A be a definable subset of 〈X〉 of positive density in a definablyamenable pair (G,X). We say that A is principal over the parameter set B if

µ(A ∩ (Y · Y )) > 0

whenever Y is a B-definable neighborhood of the identity such that finitely manyleft translates of Y cover A · A−1 ·A ·A−1.

Analogously, we say that A is hereditarily principal over the parameter set B ifall of its B-definable subsets of positive density are principal.

Remark 5.5. Let A ⊆ 〈X〉 be a definable subset of positive density of a definablyamenable pair (G,X) such that µ(A ∩ (Y · Y )) = µ(A), whenever Y is a definableneighborhood of the identity which covers A ·A−1 ·A ·A−1 with finitely many lefttranslates. Then the set A is hereditarily principal over any subset of parameters.

Proof. Let A0 be a definable subset of A of positive measure. By Remark 2.1,the set A0 does not divide. By Ehrenfeucht-Mostowski, there is a maximal finitesubset F of (AA−1)2 with the property that µ(xA0 ∩ yA0) = 0 for any two distinctx and y in F . In particular, the set (AA−1)2 is contained in FA0A

−10 . Thus, any


definable neighborhood Y of the identity such that finitely many left translates ofcover A0A

−10 A0A

−10 also cover AA−1AA−1, so µ(A∩(Y ·Y )) = µ(A) by assumption

on A. Hence µ(A0 ∩ (Y Y )) = µ(A0) > 0, as desired. �

Example 5.6. If G is generically principal, every definable subset A of positivedensity is hereditarily principal over any parameter set, since µ(Y · Y ) = 1 for anydefinable subset Y of positive measure by Lemma 4.4. By the previous remark,the definable subset A satisfies that µ(A ∩ (Y · Y )) = µ(A), so A is hereditarilyprincipal over any subset of parameters. Notice that, finitely many translates of Ycover the group G itself, whenever finitely many translates of Y cover a four-foldproduct of a definable subset of positive density.

A standard standard ultraproduct argument using Łoś’s theorem implies thefollowing: given real numbers δ > 0, ρ > 0 and η > 0, there is some d = d(δ, ρ, η)and a natural number k = k(δ, ρ, η) such that every subset A of a d-quasirandomfinite group G of density δ is hereditarily (k, 1− η)-principal up to ρ.

Principal definable sets contain weakly random principal types, whereas everyweakly random type in a hereditarily principal definable set must be principal.These notions will allow us to reproduce partially the proof of Corollary 4.8 inorder to provide a local version of [6, Theorem 5.3] to count the number of tuplessuch that all its possible products (enumerated in an increasing order) lie in a fixedhereditarily principal set of positive density. The weaker notion of principality isalready sufficient to show that the set A · B ∩ C has positive density, wheneverA, B and C are principal of positive density. We will state below the particularstatement for subsets of small tripling and leave to the reader the general statementfor principal sets in a given definably amenable pair.

Theorem 5.7. Let K > 0 and δ > 0 be given real numbers. There are real valuesǫ = ǫ(K, δ) > 0 and η = η(K, δ) > 0 as well as a natural number k = k(K, δ) suchthat for every group G and a finite subset X of G of tripling at most K together with(k, ǫ)-principal subsets A,B and C of X of relative density at least δ with respectto X, the collection of triples

{(a, b) ∈ A×B | a · b ∈ C}

has size at least η|X |2.

Proof. Assume for a contradiction that the statement does not hold. Negatingquantifiers there are positive constants K and δ such that for each triple ℓ̄ =(k, n,m) of natural numbers there exists a group Gℓ̄ and a finite subset Xℓ̄ of Gℓ̄ oftripling at most K as well as (k, 1/n)-principal subsets Aℓ̄, Bℓ̄ and Cℓ̄ of Xℓ̄, eachof relative density at least δ with respect to Xℓ̄, such that the cardinality of thesubset

Y(Gℓ̄) = {(x, y) ∈ Aℓ̄ ×Bℓ̄ | x · y ∈ Cℓ̄}

is bounded above by |Xℓ̄|2/m.

Following the approach of the Example 1.4 (b), we consider a suitable countableexpansion L of the language of groups and regard each such group Gℓ̄, with ℓ̄ ofthe form (k, k, k), as an L-structure Mℓ̄ in such a way that L contains predicatesfor Aℓ̄, Bℓ̄, Cℓ̄ and Xℓ̄. Identify now the set of such triples (k, k, k) with the naturalnumbers in a natural way and choose a non-principal ultrafilter U on N. Considerthe ultraproduct M =

∏

U Mℓ̄. As outlined in the Example 1.4, this construc-tion gives rise to a definable amenable pair (G,X) with respect to a measure µ


equipped with ∅-definable subsets A,B and C of X , each of positive density, suchthat µ2(Y(G)) = 0. Notice that A, B and C are now principal over the parameterset M , by Łoś’s theorem.

Fix a countable elementary substructure M0 of M . A straight-forward compact-ness argument yields that any countable decreasing chain of M0-definable subsetsof 〈X〉 of positive density is weakly random (as a partial type). Moreover, since theelementary substructure M0 as well as the language are countable, we can writethe type-definable subgroup 〈X〉00M0

as a countable intersection

〈X〉00M0=

⋂

i∈N

Vi,

where the decreasing chain (Vi)i∈N consists of M0-definable neighborhoods of theidentity such that Vi+1 · Vi+1 ⊆ Vi for all i in N.

Since 〈X〉00M0has bounded index in the subgroup 〈X〉, finitely many translates

of each Vi cover the subset X ·X−1 ·X ·X−1, by compactness (yet the number oftranslates possibly depends on i). The set A is principal, so A ∩ (Vi+1 · Vi+1) musthave positive density and hence so does A ∩ Vi, for every i in N. Analogously sodo B ∩ Vi and C ∩ Vi. In particular, the partial types A ∩ 〈X〉00M0

, B ∩ 〈X〉00M0and

C ∩ 〈X〉00M0are weakly random, so we can complete them to three weakly random

types p, q and r in 〈X〉00M0, containing A, B and C respectively. Corollary 3.4

applied to the triple (p, q, r) yields that the partial type {(x, y) ∈ p× q |xy ∈ r} isweakly random. Consequently the superset

Y(G) = {(x, y) ∈ A×B |xy ∈ C}

has positive density with respect to µ2, which contradicts the ultraproduct con-struction. �

In order to generalize the previous result to arbitrarily but finitely many subsets,we need to impose that the subsets are hereditary principal and not just principal.

Theorem 5.8. For a natural number n ≥ 3, let real numbers K > 0 and δF > 0,for ∅ 6= F ⊆ {1, . . . , n} be given. There are ǫ = ǫ(n,K, δF ) > 0, ρ = ρ(n,K, δF )and η = η(n,K, δF ) > 0 as well as a natural number k = k(n,K, δF ) such that forevery group G and a finite subset A of G of tripling at most K together with subsetsAF of A of relative density at least δF with respect to A such that

|{(a1, . . . , an) ∈ Gn | aF ∈ AF for all ∅ 6= F ⊆ {1, . . . , n}}| < η|A|n,

where aF stands for the product, enumerated in an increasing order, of all ai’s withi in F , then some AF cannot be hereditarily (k, ǫ)-principal up to ρ.

Since subsets of positive density have small tripling, we conclude immediatelythe following result, which relates to [14, Theorem 3.7], setting AF = A for a fixedsubset A of density at least δ containing no dense subsets avoiding products.

Corollary 5.9. Fix a natural number n ≥ 3 and let δF > 0 for ∅ 6= F ⊆ {1, . . . , n}be given. There are ǫ = ǫ(n, δF ) > 0, ρ = (n, δF ) and η = η(n, δF ) > 0 and anatural number k = k(n, δF ) such that for every finite group G and subsets AF ofG of density at least δF which are all hereditarily (k, ǫ)-principal up to ρ, we havethat

|{(a1, . . . , an) ∈ Gn | aF ∈ AF for all ∅ 6= F ⊆ {1, . . . , n}}| ≥ η|G|n.


Proof of Theorem 5.8. As in the proof of Theorem 5.7 (cf. Example 1.4 (b)) theresult follows immediately from a standard ultraproduct argument using Łoś’s theo-rem (and implicitly that a non-principal ultraproduct of finite sets is ℵ1-saturated.),together with the following claim.

Claim. In a definably amenable pair M = (G,X) with associated measure µ, con-sider definable subsets AF , for ∅ 6= F ⊆ {1, . . . , n}, of X of positive density whichare all hereditarily principal over the parameter set G. For every countable el-ementary substructure M0 such that both the measure and the sets AF ’s are allM0-definable, there is a tuple (a1, . . . , an) in Gn weakly random over M0 such thatthe product aF lies in AF for every subset F as above.

Proof of Claim. We proceed by induction on the natural number n. Since both thebase case n = 3 and the induction step have similar proofs, we will assume that thestatement of the Claim has already been shown for n− 1.

As in the proof of Theorem 5.7, the partial type AF ∩ 〈X〉00M0is weakly random

for each non-empty subset F of {1, . . . , n}. Thus, choose for every subset F aweakly random type pF over M0 in 〈X〉00M0

containing the definable subset AF .Invariance of the measure, saturation and Theorem 3.3 applied to each triple of theform (p−1

F1, p−1

1 , p−11,F1

), with ∅ 6= F1 ⊆ {2, . . . , n}, yield a realization a1 of p1 suchthat the M -definable subset

BF1= AF1

∩ a−11 A1,F1

has positive density. Notice that the set BF1is no longer definable over M0.

Downwards Löwenheim-Skolem produces a countable elementary substructureM1 containing M0 ∪ {a1}. Since AF1

is hereditarily principal over the parameterset M , so is the definable subset BF1

. By induction, we find a tuple (a2, . . . , an),weakly random over M1, such that the product aF1

lies in BF1for every subset

∅ 6= F1 ⊆ {2, . . . , n}. For n = 3, we obtain such a tuple follows by applyingTheorem 5.7 (or rather the general version behind it) to the principal M1-definablesets B2, B3 and B1,2.

Lemma 1.9 yields now that the tuple (a1, . . . , an) is weakly random over M0. Byconstruction, the product aF lies in AF for every subset ∅ 6= F ⊆ {1, . . . , n}, asdesired. �Claim

�

6. Solving equations and Roth’s theorem on progressions

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 = 2y. In fact, our methodsadapt to the non-abelian context, as well as to countable amenable group, andallow us to study more general equations such as xn · ym = zr for n +m = r. Inparticular, this yields an alternative proof to the existence in [1, Corollary 6.5] and[23, Theorem 1.2] of non-trivial solutions of the equation x · z = y2 in finite groupsof odd order, though our methods are non-quantitative.

We first introduce an auxiliary definition: Given a definably amenable pair(G,X), a definable functions f : X → 〈X〉 preserves weakly random elements iffor every element a in X and every subset C of parameters over which f is defined,


we have that a is weakly random over C if and only if f(a) is weakly random overC.

Theorem 6.1. Given a definably amenable pair (G,X) equipped with three defin-able functions fi : X → 〈X〉 for 1 ≤ i ≤ 3, each preserving weakly random elements.The set

{(x1, x2) ∈ X ×X | f1(x1) · f2(x2) ∈ f3(X)}

has positive µ2-density, whenever the equation f1(x)·f2(x) = f3(x) holds in a subsetof X of positive density.

Proof. Choose a countable elementary substructure M0 of the ambient model suchthat the measure µ as well as the definable functions fi’ s are all definable over M0.

Since the definable subset given by equation f1(x) · f2(x) = f3(x) has positivedensity, it contains a weakly random element a overM0. Now set pi = tp(fi(a)/M0)and note that each type pi lies in 〈X〉 for 1 ≤ i ≤ 3. By assumption, the types p1,p2 and p3 are again weakly random over M0. Since f1(a) · f2(a) = f3(a), the cosetsof 〈X〉00M0

of the pi’s are compatible:

CM0(p1) · CM0

(p2) = CM0(p3).

Theorem 3.3 yields a realizations b1 of p1 weakly random over M0, b2, with b2realizing p2, such that b1 · b2 belongs to f3(X). Write bi = fi(ai) for some ai in Xand notice that we can take b1 weakly random over M0, b2, a2. It then follows thata2 is weakly random over M0 and that a1 is weakly random over M0, a2 since thefunctions f1 and f2 preserve weakly random elements. As before, by Lemma 1.9the weakly random pair (a1, a2) lies in the M0-definable subset

Λ = {(x1, x2) ∈ X ×X | f1(x1) · f2(x2) ∈ f3(X)} ,

so the set Λ has positive density in X ×X with respect to the measure µ2, whichgives the desired result. �

Remark 6.2. The examples 1.4, 1.5 and 1.6 all have the property that definablefunctions with finite fibers preserve weakly random elements.

The above result specializes to the finite setting as follows:

Corollary 6.3. For every K ≥ 1 and non-zero natural numbers k and m there issome η = η(K, k,m) > 0 with the following property: Given a subset X of smalltripling K in an arbitrary group G and any three functions f1, f2 and f3 from Xto X⊙m, each with fibers of size at most k, such that

f1(x) · f2(x) = f3(x) for all x ∈ X,

then

|{(x1, x2, x3) ∈ X ×X ×X | f1(x1) · f2(x2) = f3(x3)}| ≥ η|X |2.

In particular, whenever the finite abelian group G has odd order, we deduceRoth’s theorem on the existence of 3-AP’s for subsets of small tripling, settingm = 2 and the functions f1 = f2 : x 7→ x as well as f3 : x 7→ 2x.

Proof. As in the proof of Theorem 5.7, we proceed by contradiction using Łoś’stheorem. Assuming that the statement does not hold, there are K ≥ 1 and naturalnumbers k and m such that for each n in N, we find a subset Xn of tripling K in


a group Gn, as well as functions fi,n : Xn → X⊙mn , for 1 ≤ i ≤ 3, of fibers at most

k such that

f1,n(x) · f2,n(x) = f3,n(x) for all x ∈ Xn,

yet the number of triples (x1, x2, x3) in Xn × Xn × Xn as above is bounded by|Xn|

2/n.As before, a non-principal ultrafilter on N produces an ultraproduct M in a

suitable language L which gives rise to a definable group G equipped with a dis-tinguished definable subset X such that (G,X) form a definably amenable pair asexplained in Example 1.4(b). Furthermore, we also obtain three definable functionsf1, f2 and f3 from X to X⊙m whose fibers have size at most k and such that

f1(x) · f2(x) = f3(x) for all x ∈ X.

By the Remark 6.2, each of the definable functions preserves weakly random ele-ments, since the size of each fiber is bounded uniformly. An immediate applicationof Theorem 6.1 yields the desired contradiction. �

Remark 6.4. An inspection of the proof yields that the condition f1(x) · f2(x) =f3(x) for all x in A can be replaced by the condition that

|{x ∈ X | f1(x) · f2(x) = f3(x)}| ≥ ǫ|A|

for some constant ǫ > 0 given beforehand, for this condition is sufficient to obtaina weakly random element g over M0 with f1(g) · f2(g) = f3(g).

Remark 6.5. Observe that some compatibility condition on the equation is nec-essary for the statements above to hold, as the equation x ·y = z has no solution ina product-free subset of density at least δ. Nonetheless, the strategy above permitsto find solutions for this equation in some special circumstances, such as in ultra-quasirandom groups. Another remarkable instance of solving equations in a groupis Schur’s proof [26, Hilfssatz] on the existence of a monochromatic triangle in anyfinite coloring (or cover) of the natural numbers 1, . . . , N , for N sufficiently large.In this particular case, the corresponding equation is again x · y = z. Sanders [24]remarked that Schur’s original proof can be adapted in order to count the numberof monochromatic triples (x, y, x · y). Since any weakly random type p in G00

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 Theorem 5.7 yields the following resultof Sanders [24, Theorem 1.1]:

For every natural number k ≥ 1 there is some η = η(k) > 0 with the followingproperty: Given any coloring on a finite group G with k many colors A1, . . . , Ak,there exists some color Aj, with 1 ≤ j ≤ k, such that

|{(a, b, c) ∈ Aj ×Aj ×Aj | a · b = c}| ≥ η|G|2.

Notice that the color Aj above will not be product-free, for the equation x · y = zhas a solution in Aj . For ultra-quasirandom groups, no set of positive density isproduct-free. In fact Gowers showed a stronger version [6, Theorem 5.3] of Schur’stheorem, taking AF = A for ∅ 6= F ⊆ {1, . . . , n} with the notation of Corollary 4.8.

Our attempts to provide alternative proofs of Corollary 4.8 for arbitrary ultra-products of finite groups, without assuming generic principality (or ultra-quasi-randomness), led us to isolate a particular instance of a complete amalgamationproblem (cf. the question in the Introduction).


Question. Let M0 be a countable elementary substructure of a sufficiently saturateddefinably amenable pair (G,X) and p be a weakly random type in 〈X〉00M0

. Givena natural number n, is there a tuple (a1, . . . , an) in Gn weakly random over M0

such that aF realizes p for all ∅ 6= F ⊆ {1, . . . , n}, where aF stands for the product,enumerated in an increasing order, of all ai 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, itwould imply by a standard compactness argument a finitary version of Hindman’sTheorem [9], which echoes the statement in Corollary 5.9.

Remark 6.6. If the above question has a positive answer, then for every naturalnumbers k and n there is some constant η = η(k, n) > 0 such that in any coloringon a finite group G with k many colors A1, . . . , Ak, there exists some color Aj , with1 ≤ j ≤ k such that

|{(a1, . . . , an) ∈ Gn | aF ∈ Aj for all ∅ 6= F ⊆ {1, . . . , n}}| ≥ η|G|n,

where aF stands for the product, enumerated in an increasing order, of all ai withi in F .

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doi:10.1017/fms.2013.2

Abteilung für Mathematische Logik, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, D-79104 Freiburg, Germany

Departamento de Álgebra, Facultad de Matemáticas, Universidad Complutense deMadrid, 28040 Madrid, Spain

Email address: [email protected]

Email address: [email protected]

doi:10.1017/fms.2013.2

arXiv:2009.08967v3 [math.LO] 17 Jan 2022

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