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STRUCTURE AND REGULARITY FOR SUBSETS OF GROUPS

WITH FINITE VC-DIMENSION

G. CONANT, A. PILLAY, AND C. TERRY

Abstract. Suppose G is a finite group and A ⊆ G is such that {gA : g ∈ G}has VC-dimension strictly less than k. We find algebraically well-structured

sets in G which, up to a chosen ǫ > 0, describe the structure of A and behaveregularly with respect to translates of A. For the subclass of groups withuniformly fixed finite exponent r, these algebraic objects are normal subgroupswith index bounded in terms of k, r, and ǫ. For arbitrary groups, we useBohr neighborhoods of bounded rank and width inside normal subgroups ofbounded index. Our proofs are largely model theoretic, and heavily rely on astructural analysis of compactifications of pseudofinite groups as inverse limitsof Lie groups. The introduction of Bohr neighborhoods into the nonabeliansetting uses model theoretic methods related to the work of Breuillard, Green,and Tao [8] and Hrushovski [25] on approximate groups, as well as a result ofAlekseev, Glebskiı, and Gordon [1] on approximate homomorphisms.

1. Introduction and statement of results

Szemeredi’s Regularity Lemma [47] is a fundamental result about graphs, whichhas found broad applications in graph theory, computer science, and arithmeticcombinatorics. Roughly speaking, the regularity lemma partitions large graphsinto few pieces so that almost all pairs of pieces have uniform edge density. In2005, Green [22] proved the first arithmetic regularity lemma, which uses discreteFourier analysis to define arithmetic notions of regularity for subsets of finite abeliangroups. For groups of the form Fn

p , Green’s result states that given A ⊆ Fnp , there

is H ≤ Fnp of bounded index such that A is uniformly distributed in almost all

cosets of H (as quantified by Fourier analytic methods; see [22, Theorem 2.1]).Arithmetic regularity lemmas, and their higher order analogues (see [21, 23]), arenow important tools in arithmetic combinatorics. From a general perspective, onecan view regularity lemmas as tools for decomposing mathematical objects intoingredients that are easier to study because they are either highly structured (e.g.,the cosets of a subgroup) or highly random (e.g., uniformly distributed).

Recently, a large body of work has developed around strengthened regularitylemmas for classes of graphs which forbid some particular bipartite configuration.This setting is fundamental in both combinatorics and model theory, although oftenfor very different reasons. In combinatorics, forbidden configurations can lead tosignificant quantitative improvements in results about graphs, and several well-known open problems arise in this pursuit (e.g., the Erdos-Hajnal conjecture; see 1.4of [15], and also [18]). In model theory, the focus is usually on infinite objects, andforbidden configurations are used to obtain qualitative results about definable sets

Date: September 30, 2019.2010 MSC. Primary: 03C45, 03C20, 20D60, 22C05; Secondary: 22E35.The authors were partially supported by NSF grants: DMS-1855503 (Conant); DMS-136702,

DMS-1665035, DMS-1790212 (Pillay); DMS-1855711 (Terry).

1

http://arxiv.org/abs/1802.04246v3

2 G. CONANT, A. PILLAY, AND C. TERRY

in mathematical structures. Indeed, much of modern model theory emerged fromthe study of mathematical structures in which every definable bipartite graph omitsa finite “half-graph” as an induced subgraph (such structures are called stable).

In combinatorics and model theory, the practice of forbidding finite bipartiteconfigurations is rigorously formulated using VC-dimension. By definition, the VC-dimension of a bipartite graph (V,W ;E) is the supremum of all k ∈ Z+ such that(V,W ;E) contains ([k],P([k]);∈) as an induced subgraph (where [k] = {1, . . . , k}).We call (V,W ;E) k-NIP if it has VC-dimension at most k−1.1 While this definitionis based on omitting one specific bipartite graph, an illuminating exercise is that if(V,W ;E) omits some finite bipartite graph (V ′,W ′, E′) as an induced subgraph,then (V,W ;E) is k-NIP for some k ≤ |V ′| + ⌈log2 |W

′|⌉. Therefore, having finiteVC-dimension is equivalent to omitting some finite bipartite configuration.

In [2, Lemma 1.6], Alon, Fischer, and Newman proved a strengthened regularitylemma for finite graphs of bounded VC-dimension2, in which the bound on the sizeof the partition is polynomial in the degree of irregularity (in contrast to Szemeredi’soriginal work, where these bounds are necessarily tower-type [19]), and the edgedensity in any regular pair is close to 0 or 1. This latter condition says that, as abipartite graph, each regular pair is almost empty or complete, and so the normally“random” ingredients of Szemeredi regularity are in fact highly structured. Graphregularity with bounded VC-dimension was also developed by Lovasz and Szegedyin [32], and similar results have been found for hypergraphs (e.g., [16, 17, 18]), aswell as for various model theoretic settings inside NIP (see [4, 10, 11, 34]). Thestrongest conclusion is for stable graphs3, where Malliaris and Shelah prove theexistence of regular partitions with polynomial bounds, no irregular pairs, and thesame “0-1” behavior of edge densities in regular pairs (see [34, Theorem 5.18]).

The goal of this article is to develop arithmetic regularity for arbitrary finitegroups in the context of forbidden bipartite configurations, as quantified by VC-dimension. In analogy to the case of graphs, we show that by forbidding finitebipartite configurations, one obtains a strengthened version of arithmetic regularityin which the normally random ingredients are instead highly structured. Moreover,our proof methods deepen the connection between model theory and arithmeticcombinatorics, in that we use pseudofinite methods to extend combinatorial re-sults for finite abelian groups to the nonabelian setting. This is in the same veinas Hrushovski’s [25] celebrated work on approximate groups, and the subsequentstructure theory proved by Breuillard, Green, and Tao [8]. We will use similartechniques in order to formulate arithmetic regularity in nonabelian groups usingBohr neighborhoods, which are fundamental objects from arithmetic combinatoricsin abelian groups. Finally, our results show that arithmetic regularity for NIP setsin finite groups coincides with a certain model theoretic phenomenon called “com-pact domination”. This notion was first isolated by Hrushovski, Peterzil, and thesecond author [26] in their proof of the so-called “Pillay conjectures” for groupsdefinable in o-minimal theories, and later played an important role in the study ofdefinably amenable groups definable in NIP theories [9, 26, 28, 29].

1This terminology is from model theory, where a bipartite graph with infinite VC-dimensionis said to have the independence property, and so NIP stands for “no independence property”.

2The VC-dimension of a graph (V ;E) is that of its “bipartite double cover” (V, V ;E).3A bipartite graph is called k-stable if it omits ([k], [k];≤) as an induced subgraph; and a

graph is k-stable if its bipartite double cover is. Note that a k-stable bipartite graph is k-NIP.

STRUCTURE AND REGULARITY FOR VC-SETS IN GROUPS 3

Before stating the main results of this paper, we briefly recall previous workon stable arithmetic regularity, as it provides a template for the “structure andregularity” statements we will obtain in the NIP setting. Given a group G and asubset A ⊆ G, we define the bipartite graph ΓG(A) = (V,W ;E) where V = W = Gand E = {(x, y) ∈ G2 : yx ∈ A}.4 Given k ≥ 1, we say that a subset A of a groupG is k-NIP (respectively, k-stable) if ΓG(A) is k-NIP (respectively, k-stable), asdefined above. In [49], the third author and Wolf developed arithmetic regularityfor k-stable subsets of Fn

p . They prove that such sets satisfy a strengthened versionof Green’s arithmetic regularity lemma above, in which there is an efficient boundon the index of H and A is uniformly distributed in all cosets of H . They alsoshow that a k-stable subset of Fn

p is approximately a union of cosets of a subgroupof small index, which is an arithmetic analogue of “0-1 density” in regular pairs.In [14, Theorem 1.2], we generalized and strengthened the results from [49] on Fn

p

to the setting of arbitrary finite groups, but without explicit bounds.5

Theorem 1.1. [14] For any k ≥ 1 and ǫ > 0, there is n = n(k, ǫ) such that thefollowing holds. Suppose G is a finite group and A ⊆ G is k-stable. Then there isa normal subgroup H ≤ G, of index at most n, satisfying the following properties.

(i) (structure) There is a set D ⊆ G, which is a union of cosets of H, such that

|A△D| < ǫ|H |.

(ii) (regularity) For any g ∈ G, either |gH ∩ A| < ǫ|H | or |gH\A| < ǫ|H |.

Moreover, H is in the Boolean algebra generated by {gAh : g, h ∈ G}.

Our first result on arithmetic regularity in the setting of bounded VC-dimensionis for k-NIP subsets of finite groups with uniformly bounded exponent.

Theorem 3.2. For any k, r ≥ 1 and ǫ > 0, there is n = n(k, r, ǫ) such that thefollowing holds. Suppose G is a finite group of exponent r, and A ⊆ G is k-NIP.Then there are

∗ a normal subgroup H ≤ G of index at most n, and∗ a set Z ⊆ G, which is a union of cosets of H with |Z| < ǫ|G|,

satisfying the following properties.


|(A\Z)△D| < ǫ|H |.

(ii) (regularity) For any g ∈ G\Z, either |gH ∩A| < ǫ|H | or |gH\A| < ǫ|H |.


Thus the behavior of NIP sets in bounded exponent groups is almost identical tothat of stable sets in arbitrary finite groups, where the only difference is the errorset Z. This reflects similar behavior in graph regularity, where the main differencebetween the stable and NIP cases is the need for irregular pairs. Theorem 3.2 alsoqualitatively generalizes and strengthens a recent quantitative result of Alon, Fox,and Zhao [3, Theorem 1.1] on k-NIP subsets of finite abelian groups of uniformly

4This is a bipartite analogue of the “Cayley sum-graph of A in G”, as defined in [49] for abeliangroups. The reason we use yx ∈ A rather than xy ∈ A is due to the model theoretic preferencefor “left-invariant” formulas (see Section 2.4 for details).

5Quantitative results for stable sets in finite abelian groups were later proved by the thirdauthor and Wolf (see [50, Theorem 4]). A quantitative analogue of Theorem 1.1 remains open.


bounded exponent. A version of Theorem 3.2 with polynomial bounds (in ǫ-1), butslightly weaker qualitative ingredients, is conjectured in [3].6

We then turn to k-NIP sets in arbitrary finite groups. In this case, one cannotexpect a statement involving only subgroups, as in Theorem 3.2. Indeed, as notedin [3, Section 5], if p ≥ 3 is prime and A = {1, 2, . . . , ⌊p2⌋}, then A is 4-NIP as asubset of Z/pZ, but A cannot be approximated as in Theorem 3.2 for arbitrarilysmall ǫ. This example illustrates a common obstacle faced in arithmetic combina-torics when working in abelian groups with very few subgroups. In situations likethis, one often works instead with certain well-structured subsets of groups calledBohr neighborhoods. Bohr neighborhoods in cyclic groups were used in Bourgain’simprovement of Roth’s Theorem [7, (0.11)], and also form the basis of Green’sarithmetic regularity lemma for finite abelian groups [22, Theorem 5.2]. Resultsrelated to ours involving Bohr neighborhoods, with quantitative bounds but forabelian groups, were independently obtained by Sisask [46, Theorem 1.4].

Given a group H , an integer r ≥ 0, and some δ > 0, we define a (δ, r)-Bohrneighborhood in H to be a set of the form Br

τ,δ := {x ∈ H : d(τ(x), 0) < δ},where τ : H → Tr is a group homomorphism and d is a fixed invariant metric on ther-dimensional torus Tr (see Remark 4.6 and Definition 4.7). Our main structureand regularity result for NIP sets in finite groups is as follows.

Theorem 5.7. For any k ≥ 1 and ǫ > 0 there is n = n(k, ǫ) such that the followingholds. Suppose G is a finite group and A ⊆ G is k-NIP. Then there are

∗ a normal subgroup H ≤ G of index m ≤ n,∗ a (δ, r)-Bohr neighborhood B in H, where 0 ≤ r ≤ n and 1

n ≤ δ ≤ 1, and∗ a subset Z ⊆ G, with |Z| < ǫ|G|,


(i) (structure) There is a set D ⊆ G, which is a union of at most m(2δ )r translatesof B, such that

|(A△D)\Z| < ǫ|B|.

(ii) (regularity) For any g ∈ G\Z, either |gB ∩ A| < ǫ|B| or |gB\A| < ǫ|B|.

Moreover, H and Z are in the Boolean algebra generated by {gAh : g, h ∈ G}, andif G is abelian then we may assume H = G.

In order to prove Theorems 3.2 and 5.7, we will first prove companion theoremsfor these results involving definable sets in infinite pseudofinite groups (Theorems3.1 and 5.5, respectively). We then prove the theorems about finite groups bytaking ultraproducts of counterexamples in order to obtain infinite pseudofinitegroups contradicting the companion theorems. To prove the companion theorems,we work with a saturated pseudofinite group G, and an invariant NIP formulaθ(x; y) (see Definitions 2.1 and 2.10). In [13], the first two authors proved “genericcompact domination” for the quotient group G/G00

θr , where θr(x; y, u) := θ(x · u; y)and G00

θr is the intersection of all θr-type-definable bounded-index subgroups ofG (see Definition 2.12). In this case, G/G00

θr is a compact Hausdorff group, andgeneric compact domination roughly states that if A ⊆ G is θr-definable, then theset of cosets of G00

θr , which intersect both A and G\A in “large” sets with respectto the pseudofinite counting measure, has Haar measure 0 (see Theorem 2.14).This is essentially a regularity statement for A with respect to the subgroup G00

θr ,

6Poly-exponential bounds were later proved by the first author (see [12, Theorem 1.6]).


which is rather remarkable as generic compact domination originated in [26] towardproving conjectures of the second author on the Lie structure of groups definablein o-minimal theories [39, Conjecture 1.1].

For G and θ(x; y) as above, the regularity provided by generic compact domi-nation for θr-definable sets in G cannot be transferred directly to finite groups, asthe statement depends entirely on type-definable data (such as G00

θr ). Thus, muchof the work in this paper focuses on obtaining definable approximations to G00

θr andthe other objects involved in generic compact domination. We first investigate thesituation when G/G00

θr is a profinite group, in which case G00θr can be approximated

by definable finite-index subgroups of G. Using this, we prove Theorem 3.1 (thepseudofinite companion to Theorem 3.2 above). The connection to Theorem 3.2is that if G is elementarily equivalent to an ultraproduct of groups of uniformlybounded exponent, then G/G00

θr is a compact Hausdorff group of finite exponent,hence is profinite (see Fact 2.3(d)).

When G/G00θr is not profinite, there are not enough definable finite-index sub-

groups available to describe G00θr , and it is for this reason that we turn to Bohr neigh-

borhoods. This is somewhat surprising, as Bohr neighborhoods are fundamentallylinked to abelian groups, and we do not make any assumptions of commutativity.However, by a general result of the second author on “definable” compactificationsof pseudofinite groups, we in fact have that the connected component of G/G00

θr isabelian (see Theorem 2.7). It is at this point that we see the beautiful partner-ship between pseudofinite groups and NIP formulas. Specifically, we have genericcompact domination of θr-definable sets by the abelian-by-profinite group G/G00

θr ,which allows us to analyze θr-definable sets in G using Bohr neighborhoods in defin-able finite-index subgroups. In order to obtain a statement involving only definableobjects, we use approximate homomorphisms to formulate a notion of approximateBohr neighborhoods. This leads to Theorem 5.5 (the pseudofinite companion ofTheorem 5.7 above). We then apply a result of Alekseev, Glebskiı, and Gordon [1,Theorem 5.13] on approximate homomorphisms to find actual Bohr neighborhoodsinside approximate Bohr neighborhoods and, ultimately, prove Theorem 5.7.

In Section 6, we prove similar results for fsg groups definable in distal NIPtheories (see Theorems 6.7 and 6.6). This follows our theme, as such groups satisfya strong form of compact domination (see Lemma 6.5). In Section 7, we discusscompact p-adic analytic groups as one concrete example of the distal fsg setting.

Acknowledgements. Much of the work in this paper was carried out during the2018 Model Theory, Combinatorics and Valued Fields trimester program at InstitutHenri Poincare. We thank IHP for their hospitality. We also thank Julia Wolf forhelpful conversations, and the anonymous referee for their careful reading and forproviding many valuable revisions, corrections, and suggestions.

2. Preliminaries

2.1. First-order structures and definability. We start by establishing the set-ting involving first-order structures. See [35] for an introduction to first-order logicand model theory.

Let L be a first-order language. Following model-theoretic convention, we willsay that an L-structure M∗ is sufficiently saturated if M∗ is κ-saturated andstrongly κ-homogeneous for some large (e.g., strongly inaccessible) cardinal κ. LetM∗ be a fixed sufficiently saturated L-structure. We say that a set is bounded if


its cardinality is strictly less than the saturation cardinal κ of M∗.7 When L andM∗ are fixed, we will refer to an L-formula with parameters from M∗ as simply aformula. We also call an elementary substructure M ≺ M∗ small if its universeis bounded. As usual, a subset X ⊆ (M∗)n is definable if there is a formulaφ(x1, . . . , xn) such that X = φ(M∗) := {a ∈ (M∗)n : M∗ |= φ(a)}; and X ⊆ (M∗)n

is type-definable if it is an intersection of a bounded number of definable subsetsof (M∗)n. We will further say that X ⊆ (M∗)n is countably-definable if it is anintersection of countably many definable subsets of (M∗)n.

Many of our results will operate in a “local” model-theoretic setting, which is tosay that we focus on a single formula θ(x; y) whose free variables are partitionedinto a singleton x and a finite tuple y.

Definition 2.1. Let θ(x; y) be a formula.

(1) A 〈θ〉-formula is a formula φ(x; y1, . . . , yn) obtained as a Boolean combi-nation of θ(x; y1), . . . , θ(x; yn) for some n ≥ 1 and variables y1, . . . , yn.

(2) An instance of θ(x; y) is a formula of the form θ(x; b) for some b ∈ (M∗)|y|.(3) A θ-formula φ(x) is an instance of a 〈θ〉-formula φ(x; y1, . . . , yn).(4) A set X ⊆M∗ is θ-definable if X = φ(M∗) for some θ-formula φ(x).(5) A setX ⊆M∗ is θ-type-definable (respectively, θ-countably-definable)

if it is an intersection of a bounded (respectively, countable) number of θ-definable subsets of M∗.

In Sections 3 and 5, we will assume L is an expansion of the language of groups8

and M∗ expands a group. So in this case, we use G in place of M∗, and just saythat G is a sufficiently saturated L-structure expanding a group.

Definition 2.2. Let G be a sufficiently saturated L-structure expanding a group.A formula θ(x; y) is (left) invariant if, for any a ∈ G and b ∈ G|y|, there is c ∈ G|y|

such that θ(a · x; b) and θ(x; c) define the same subset of G.

The typical example of an invariant formula is something of the form θ(x; y) :=φ(y · x), where φ(x) is a formula in one variable. Note also that if θ(x; y) is aninvariant formula then any 〈θ〉-formula is invariant as well.

2.2. Compact quotients. By convention, when we say that a topological spaceis compact, we mean compact and Hausdorff. We will frequently use the fact thata compact space is second-countable if and only if it is separable and metrizable(e.g., by Urysohn’s Metrization Theorem). Given a topological group K, we let K0

denote the connected component of the identity in K, which is a closed normalsubgroup of K. A topological group is called profinite if it is a projective limitof finite groups. By a Lie group, we mean a finite-dimensional real Lie group.For n ∈ N, let Tn denote the n-dimensional torus, where T = R/Z (so T0 is thetrivial group). We view Tn as a compact topological group (with the product ofthe quotient topology on R/Z). We use ∼= for isomorphism of topological groups.

The structure theory for compact groups as projective limits of compact Liegroups will play a significant role in this paper. We refer the reader to [41, Section1.1] for details on projective limits of topological spaces. Each projective limit

7This is not to be confused with later uses of the phrase “uniformly bounded” in the contextof theorems about finite groups.

8In fact, the group language expanded by a single unary relation symbol A suffices for ourmain results on finite groups (Theorems 3.2 and 5.7).


below will be indexed by a join semilattice (jsl), i.e., a poset I = (I,≤) such thatany finite set F ⊆ I has a least upper bound (denoted supF ). When we refer to N

as a jsl, we always use the usual ordering.The following are several important facts about compact groups that we will

need at various points throughout the paper.

Fact 2.3.

(a) If K and L are compact groups, and π : K → L is a surjective continuoushomomorphism, then π(K0) = L0.

(b) A compact group K is profinite if and only if K0 is trivial.(c) Any compact second-countable group admits a bi-invariant compatible metric.(d) Any compact torsion group is profinite.(e) Any compact group K admits a unique left-invariant regular Borel probability

measure ηK (called the normalized Haar measure on K).(f) If K is a compact abelian Lie group, then K ∼= Tn×F for some finite group F

and some n ∈ N.(g) If K is a compact group then there is a jsl I and a projective system (Li)i∈I of

compact Lie groups such that K ∼= lim←−

Li. Moreover, the projection maps aresurjective, and if K is second-countable then we may assume I = N.

Proof. Part (a). This is a standard exercise, which follows from compactness andthe fact that a continuous surjective homomorphism of compact groups is closed(by an easy exercise) and open (see, e.g., [24, p. 669]).

Part (b). This follows from the fact that a topological group is profinite if andonly if it is compact and totally disconnected (see [41, Theorem 2.1.3]).

Part (c). See [24, Corollary A4.19].Part (d). See [30, Theorem 4.5] (where torsion groups are called periodic).Part (e). See [24, Theorem 2.8].Part (f). See [24, Proposition 2.42].Part (g). This is part of the well-known Peter-Weyl Theorem. See [24, Corollary

2.43] for the first statement and surjectivity of the projection maps. The jsl inquestion is the collection I of kernels of continuous homomorphisms from K tounitary groups under reverse inclusion (see [24, Corollary 2.36] and its proof). Inparticular, any open neighborhood of the identity in K contains some kernel in I(see [24, Exercise E9.1]), which yields the final claim on second-countability. �

We now return to model theory. Let L be a first-order language.

Fact 2.4. Let G be a group definable in a sufficiently saturated L-structure M∗.Suppose Γ ≤ G is a type-definable normal subgroup of bounded index, and letπ : G→ G/Γ be the canonical homomorphism.

(a) G/Γ is a compact topological group, where a set X ⊆ G/Γ is closed if and onlyif π-1(X) is type-definable.

(b) If X ⊆ G is definable then the set {aΓ ∈ G/Γ : aΓ ⊆ X} is open in G/Γ.(c) G/Γ is second-countable if and only if Γ is countably-definable.(d) Suppose G = M∗ is an expansion of a group, and Γ is θ-type-definable for some

invariant formula θ(x; y). Then X ⊆ G/Γ is closed if and only if π-1(X) isθ-type-definable.

(e) Suppose µ is a left-invariant finitely additive probability measure on the Booleanalgebra B of definable subsets of G, and X is a closed subset of G/Γ. ThenηG/Γ(X) = inf{µ(Z) : Z ∈ B and π-1(X) ⊆ Z}.


Proof. Part (a). See [39, Lemma 2.7].Part (b). See [39, Remark 2.4].Part (c). See [31, Fact 1.3] for the right-to-left direction. Conversely, suppose

G/Γ is second-countable and let {Un : n ∈ N} be a neighborhood basis of theidentity in G/Γ. For any n ∈ N, π-1(Un) is co-type-definable and contains Γ. Bysaturation, there is some definable set Dn ⊆ G such that Γ ⊆ Dn ⊆ π-1(Un). Itfollows that Γ =

⋂∞n=0Dn.

Part (d). See [13, Corollary 4.2].9

Part (e). Let SG(M∗) be the space of types concentrating10 on G. Recall thatSG(M∗) is a Stone space, with a basis of clopen sets [Z] := {p ∈ SG(M) : Z ∈ p}where Z ∈ B. Note that G acts on SG(M∗) by left multiplication. Moreover,there is a unique left-invariant regular Borel probability measure µ on SG(M∗)such that µ(Z) = µ([Z]) for any Z ∈ B (see, e.g., [44, Section 7.1]). In particular,if C ⊆ SG(M∗) is closed then µ(C) = inf{µ(Z) : Z ∈ B and C ⊆ [Z]}. Nowdefine the function f : SG(M∗)→ G/Γ where f(p) is the unique coset aΓ such thatp |= aΓ. It is straightforward to show that f -1(X) = {p ∈ SG(M∗) : p |= π-1(X)}for any closed X ⊆ G/Γ. It follows that f is continuous and (using saturation ofG) that µ(f -1(X)) = inf{µ(Z) : Z ∈ B and π-1(X) ⊆ Z} for any closed X ⊆ G/Γ.Finally, one checks that µ ◦ f -1 is a left-invariant regular Borel probability measureon G/Γ, and thus must be ηG/Γ by Fact 2.3(e). �

The topology defined on G/Γ in the previous fact is called the logic topology,and the canonical homomorphism from G to G/Γ connects topological structure inG/Γ to definable sets in G. This is a special case of the following notion.

Definition 2.5. Suppose M∗ is a sufficiently saturated L-structure, X ⊆ M∗ isdefinable, Y is a compact space, and f : X → Y is a function. Then f is definableif f -1(C) is type-definable for any closed C ⊆ Y . If X is θ-definable and each f -1(C)is θ-type-definable, for some fixed formula θ(x; y), then we say f is θ-definable.

Remark 2.6. Suppose f : X → Y is definable (as above) and C ⊆ U ⊆ Y withC closed and U open. Then, by saturation of M∗, there is a definable set D ⊆ Xsuch that f -1(C) ⊆ D ⊆ f -1(U). Thus if C ⊆ Y is clopen then f -1(C) is definable.Moreover, the analogous statements hold if f is θ-definable for some formula θ(x; y).

Now let L be an expansion of the group language and suppose G is a sufficientlysaturated L-structure expanding a group. We say that G is pseudofinite if it isan elementary extension of an ultraproduct

∏U Gi, where each Gi is a finite L-

structure expanding a group and U is a (nonprincipal) ultrafilter on some index setI. The next result of the second author is the key ingredient which will allow us tointroduce Bohr neighborhoods into the setting of possibly nonabelian finite groups.

Theorem 2.7. [40] Suppose G is a sufficiently saturated pseudofinite expansion ofa group, and Γ ≤ G is normal and type-definable of bounded index. Then (G/Γ)0

is abelian.

Proof. SupposeM ≺ G is a small substructure such that Γ is type-definable overM .Then G00

M ≤ Γ, where G00M denotes the intersection of all bounded-index subgroups

9The cited result works with a formula of the form θr(x; y, u) := θ(x · u; y), where θ(x; y) isinvariant. The proof for θ(x; y) is the same.

10A type p concentrates on a type-definable set X, written p |= X, if p contains a boundedset {φi(x)}i∈I of formulas such that X =

⋂i∈I φi(M

∗).


of G that are type-definable over M . By [40, Theorem 2.2], (G/G00M )0 is abelian,

and so (G/Γ)0 is abelian by Fact 2.3(a) and since the canonical homomorphismfrom G/G00

M to G/Γ is surjective and continuous. �

Remark 2.8. The proof of [40, Theorem 2.2] uses the classification of approximategroups due to Breuillard, Green, and Tao [8]. This theorem was later generalizedby Nikolov, Schneider, and Thom [37, Theorem 8], who showed that the connectedcomponent of any compactification of an (abstract) pseudofinite group is abelian.Their proof uses the classification of finite simple groups.

We are now ready to collect all of the previous facts and prove the main result ofthis subsection. In particular, let G be a group definable in a sufficiently saturatedstructure M∗, and let Γ ≤ G be type-definable and normal of bounded index. SinceG/Γ is a compact group, and the topology on G/Γ is controlled by type-definableobjects in G, we can analyze Γ using the structure theory for compact Lie groups.The next lemma describes the main ingredients of this analysis. Given a poset I,we say that a net (Xi)i∈I of subsets of some fixed set X is decreasing if Xj ⊆ Xi

for any i, j ∈ I such that i ≤ j.

Lemma 2.9. Let G be a group definable in a sufficiently saturated L-structureM∗. Suppose Γ ≤ G is type-definable and normal of bounded index. Then thereis a bounded jsl I, a decreasing net (Γi)i∈I of countably-definable bounded-indexnormal subgroups of G, and decreasing net (Hi)i∈I of definable finite-index normalsubgroups of G, such that the following properties are satisfied.

(i) Γ =⋂

i∈I Γi, and Γi ≤ Hi for all i ∈ I.(ii) For all i ∈ I, G/Γi is a compact Lie group and Hi/Γi = (G/Γi)

0.(iii) G/Γ ∼= lim

←−G/Γi and (G/Γ)0 ∼= lim

←−Hi/Γi.

11

Moreover:

(a) If Γ is countably-definable then we may assume I = N.(b) If G = M∗ is an expansion of a group, and Γ is θ-type-definable for some

invariant formula θ(x; y), then we may assume that for all i ∈ I, Hi is θ-definable and Γi is θ-type-definable.

(c) If G/Γ is profinite then we may assume Hi = Γi for all i ∈ I.(d) If G = M∗ is a pseudofinite expansion of a group then we may assume that for

all i ∈ I, there is some ni ∈ N such that Hi/Γi∼= Tni .

(e) If G is abelian then we may assume that, for all i ∈ I, there is some ni ∈ N

and some finite group Fi such that G/Γi∼= Tni × Fi (and so Hi/Γi

∼= Tni).

Proof. By Fact 2.3(g), there is a projective system (Li)i∈I of compact Lie groupssuch that G/Γ ∼= lim

←−Li and the projection maps fi : G/Γ → Li are surjective.

Since Γ has bounded index in G, we may assume I is bounded. For i ∈ I, setΓi = ker(fi ◦ π), where π : G → G/Γ is the canonical homomorphism. Then eachΓi is a type-definable bounded-index normal subgroup of G, and Γ =

⋂i∈I Γi.

Moreover, G/Γi∼= Li for all i ∈ I, and so G/Γ ∼= lim

←−G/Γi with canonical (surjec-

tive) projection maps τi : G/Γ → G/Γi. Since each G/Γi is a Lie group (and thussecond-countable), each Γi is countably-definable by Fact 2.4(c).

Now, given i ∈ I, let Hi ≤ G be the pullback of (G/Γi)0 under G → G/Γi

(so (G/Γi)0 = Hi/Γi). Then for any i ∈ I, Hi/Γi is a clopen finite-index normal

11Here we view (G/Γi)i∈I and (Hi/Γi)i∈I as projective systems using (i) and the assumptionthat (Hi)i∈I and (Γi)i∈I are decreasing.


subgroup of G/Γi.12 So each Hi is a finite-index normal subgroup of G and, since

G → G/Γi is definable, Hi is definable by Remark 2.6. If i, j ∈ I and i ≤ j, thenthe pullback of Hi/Γi to G/Γj is a clopen finite-index normal subgroup, and thuscontains Hj/Γj, which implies Hj ≤ Hi. Finally, τi((G/Γ)0) = Hi/Γi for all i ∈ Iby Fact 2.3(a), and so (G/Γ)0 ∼= lim

←−Hi/Γi by [41, Corollary 1.1.8]. This finishes

the proof of claims (i) through (iii)Now we deal with the remaining claims. Claim (a) follows from Facts 2.3(g) and

2.4(c), claim (b) follows from Fact 2.4(d), and claim (e) follows from Fact 2.3(f).For claim (c), if G/Γ is profinite then each group Hi/Γi is trivial by Fact 2.3(a, b).Finally, for claim (d), suppose G = M∗ is a pseudofinite expansion of a group. Then(G/Γ)0 is abelian by Theorem 2.7. So, for any i ∈ I, Hi/Γi is a compact connectedabelian Lie group, and thus Hi/Γi

∼= Tni for some ni ∈ N by Fact 2.3(f). �

2.3. NIP formulas in pseudofinite groups. In this subsection, we assume thatL expands the group language. Let G be a sufficiently saturated L-structure ex-panding a group.

Definition 2.10. Let θ(x; y) be a formula. Given k ≥ 1, θ(x; y) is k-NIP if theredo not exist sequences (ai)i∈[k] in G and (bI)I⊆[k] in Gy such that θ(ai, bI) holds ifand only if i ∈ I. We say θ(x; y) is NIP if it is k-NIP for some k ≥ 1.

Remark 2.11. Given k ≥ 1, a formula θ(x; y) is k-NIP if and only if the set system{θ(G; b) : b ∈ G|y|} on G has VC-dimension at most k − 1 (see, e.g., [44, Section6.1] for details on set systems and VC-dimension).

Next, we summarize several main results from [13], which will form the basis forour work on NIP sets in finite and pseudofinite groups. We say that a set A ⊆ G isgeneric if G is covered by finitely many left translates of A; and a formula φ(x) isgeneric if φ(G) is generic. Given a formula θ(x; y), let Sθ(G) denote the space ofcomplete θ-types over G (i.e., ultrafilters over the Boolean algebra of θ-formulas).A type p ∈ Sθ(G) is generic if every formula in p is generic.

Definition 2.12. Let G be a sufficiently saturated expansion of a group. Givena formula θ(x; y), let θr(x; y, u) denote the formula θ(x · u; y), and let G00

θr be theintersection of all θr-type-definable bounded-index subgroups of G.

Note that if θ(x; y) is invariant, then so is θr(x; y, u). One can also show that ifθ(x; y) is invariant and NIP, and φ(x) is a θr-formula, then φ(y · x) is NIP.13 Wenow focus on the case that G ≻

∏U Gi is pseudofinite (as defined before Theorem

2.7). In this case, µ will always denote the pseudofinite counting measure onG, which is obtained by lifting (e.g., as in [26, Section 2]) the measure limU µi on∏

U Gi, where µi is the normalized counting measure on Gi.

Remark 2.13. In several proofs, we will apply Los’s Theorem to properties of µ,which requires an expanded language L+ containing a sort for the ordered interval[0, 1] with a distance function, and functions from the G-sort to [0, 1] giving themeasures of L-formulas. There are many accounts of this kind of formalism, andso we will omit further details and refer the reader to similar treatments in theliterature, for example [13, Section 2.2], [25, Section 2.6], and [26, Section 2].

12If K is a Lie group then it is locally connected, and so K0 is clopen. Thus, if K is alsocompact, then K0 has finite index.

13However, θr(x; y, u) itself need not be NIP (see [13, Example 3.7]).


Theorem 2.14. [13] Let G be a sufficiently saturated pseudofinite expansion of agroup, and suppose θ(x; y) is an invariant NIP formula.

(a) A θr-formula φ(x) is generic if and only if µ(φ(x)) > 0.(b) There are generic θr-types in Sθr(G).(c) G00

θr is θr-countably-definable and normal of bounded index.(d) Suppose A ⊆ G is θr-definable, and let E ⊆ G/G00

θr be the set of C ∈ G/G00θr

such that p |= C ∩A and q |= C ∩ (G\A) for some generic p, q ∈ Sθr(G). ThenE is closed and ηG/G00

θr(E) = 0.

Proof. Part (a) is [13, Corollary 2.14], and part (b) is [13, Proposition 3.12(e)].Part (b). By [13, Theorem 3.15], G00

θr is θr-type-definable of bounded index. NotethatG00

θr is ∅-invariant and, without loss of generality, L is countable. By saturation,it follows that G00

θr is countably-definable, and thus θr-countably-definable.Part (c) is [13, Theorem 6.2]. �

Remark 2.15. In the setting of the previous fact, part (a) implies that if X ⊆ G isθr-type-definable, then p |= X for some generic p ∈ Sθr(G) if and only if µ(W ) > 0for any definable (equivalently, any θr-definable) set W ⊆ G containing X .

We now prove the main result of this subsection.

Lemma 2.16. Let G be a sufficiently saturated pseudofinite expansion of a group.Suppose θ(x; y) is an invariant NIP formula, and let (Wi)

∞i=0 be a decreasing se-

quence of definable sets such that G00θr =

⋂∞i=0Wi. Fix a θr-definable set A ⊆ G.

Then, for any ǫ > 0, there is a θr-definable set Z ⊆ G and some i ∈ N such thatµ(Z) < ǫ and, for any g ∈ G\Z, either µ(gWi ∩ A) = 0 or µ(gWi\A) = 0.

Proof. To ease notation, let Γ = G00θr and K = G/Γ, and let π : G → K be the

canonical homomorphism. Let E be as in Theorem 2.14(d) (with respect to thefixed θr-definable set A). So E is closed and ηK(E) = 0.

Fix ǫ > 0. By Fact 2.4(d, e), we may fix a θr-definable set Z ⊆ G such thatµ(Z) < ǫ and π-1(E) ⊆ Z. We show that Z satisfies the conclusion of the lemma.To motivate the argument, we first make a side remark. In particular, by saturationand Remark 2.15, we immediately have that for any g ∈ G\Z, there is some i ∈ N

such that µ(gWi ∩ A) = 0 or µ(gWi\A) = 0. Thus the content of the followingargument is to show that we can pick one i ∈ N that works for every g ∈ G\Z.

Toward a contradiction, suppose that for all i ∈ N there is some ai ∈ G\Z suchthat µ(aiWi∩A) > 0 and µ(aiWi\A) > 0. Let U = {C ∈ K : C ⊆ Z}, which is openin K by Fact 2.4(b). Note that E ⊆ U and π-1(U) ⊆ Z. In particular, (aiΓ)∞i=0 is aninfinite sequence in K\U . Since U is open and K is compact and second-countable(by Fact 2.4(c) and Theorem 2.14(c)), we may pass to a subsequence and assumethat (aiΓ)∞i=0 converges to some aΓ ∈ K\U . In particular, aΓ 6∈ E.

Claim: For all i ∈ N, µ(aWi ∩ A) > 0 and µ(aWi\A) > 0.Proof : First, given i ∈ N, let Ui = {C ∈ K : C ⊆ aWi}. As above, each Ui is openin K. Moreover, for any i ∈ N, since Γ ⊆ Wi, we have aΓ ∈ Ui ⊆ aWi/Γ. Also,since (Wi)

∞i=0 is decreasing and Γ =

⋂∞i=0Wi is a subgroup of G, it follows from

saturation that, for all i ∈ N, there is ni ∈ N such that WniWni

⊆Wi.Now fix i ∈ N. Since Uni

is an open neighborhood of aΓ, there is j ≥ ni suchthat ajΓ ∈ Uni

. In particular, aj ∈ π-1(Uni) ⊆ aWni

. Now we have ajWj ⊆ajWni

⊆ aWniWni

⊆ aWi. Since µ(ajWj ∩ A) > 0 and µ(ajWj\A) > 0, we haveµ(aWi ∩ A) > 0 and µ(aWi\A) > 0. ⊣


Now, since (Wi)∞i=0 is decreasing and Γ =

⋂∞i=0Wi, it follows from saturation

that any definable set containing aΓ ∩ A (respectively, aΓ\A) contains aWi ∩ A(respectively, aWi\A) for some i ∈ N. So aΓ ∈ E by the claim (and Remark 2.15),which is a contradiction. �

Note that Lemma 2.16 provides a “regularity” statement for invariant NIP for-mulas in pseudofinite groups. However, it does not provide any information aboutthe shape of the definable sets Wi, other than that they approximate the subgroupG00

θr . Indeed, most of the remaining work in this paper is toward replacing thesedefinable sets with ones that enjoy meaningful algebraic properties.

Remark 2.17. Except for Sections 6 and 7, our use of finite VC-dimension isbased entirely on applications of Lemma 2.16. So it is worth emphasizing that theproof of Theorem 2.14 heavily uses fundamental results about VC-dimension suchas the Sauer-Shelah Lemma, the VC-Theorem, and Matousek’s general version ofthe (p, q)-theorem [36, Theorem 4]. See also [9] and [44, Chapter 6].

2.4. NIP subsets of groups. In this subsection, we briefly recall the notion of anNIP subset of an arbitrary group, and clarify the relationship to VC-dimension andNIP formulas. Recall from the introduction that if G is a group and A ⊆ G, thenwe have the bipartite graph ΓG(A) = (G,G;E) where E = {(x, y) ∈ G2 : yx ∈ A}.

Definition 2.18. Given a group G and an integer k ≥ 1, we say that A is k-NIP(in G) if ΓG(A) is k-NIP, i.e., it omits the bipartite graph ([k],P([k]);∈) as aninduced subgraph (where [k] = {1, . . . , k}).

Remark 2.19. Fix a group G and a subset A ⊆ G.

(1) If ΓG(A) omits some finite bipartite graph (V,W,E) as an induced sub-graph, then A is k-NIP for some k ≤ |V |+ ⌈log2 |W |⌉.

(2) Given k ≥ 1, A is k-NIP if and only if the formula θ(x; y) := A(y · x) isk-NIP (here we view G as a structure in the group language expanded bya predicate for A).

(3) Given k ≥ 1, A is k-NIP if and only if the set system {gA : g ∈ G} of lefttranslates of A has VC-dimension at most k − 1 (see Remark 2.11).

Note that the formula θ(x; y) defined in the second remark is invariant. Thisis the main reason we define ΓG(A) in terms of the edge relation yx ∈ A and notxy ∈ A. However, an important fact is that if a bipartite graph (V,W ;E) is k-NIPthen the “opposite graph” (W,V ; {(w, v) : E(v, w)}) is 2k-NIP (see [44, Lemma6.3]).14 So the order of the group operation when defining k-NIP sets only affectsthe precise value of k, and not whether the set is NIP overall.

3. Structure and regularity: the profinite case

In this section, we prove a structure and regularity theorem for θr-definable setsin a sufficiently saturated pseudofinite group G, where θ(x; y) is an invariant NIPformula andG/G00

θr is profinite (Theorem 3.1 below). As an application, we obtain astructure and regularity theorem for NIP sets in finite groups of uniformly boundedexponent. We also view Theorem 3.1 as a warm-up to the general case. Indeed, this

14On the other hand, if a bipartite graph is k-stable (as defined in the introduction) then onecan easily check that the opposite graph is as well. This reconciles our definitions with those in[14], where we defined k-stable subsets of groups using the relation xy ∈ A.


theorem follows almost immediately from Lemma 2.16 and the fact that profinitequotients correspond to type-definable subgroups that are intersections of definablesubgroups (via Lemma 2.9(c)).

Theorem 3.1. Let G be a sufficiently saturated pseudofinite expansion of a group,and suppose θ(x; y) is an invariant NIP formula. Assume G/G00

θr is profinite. Fixa θr-definable set A ⊆ G and some ǫ > 0. Then there are

∗ a θr-definable finite-index normal subgroup H ≤ G, and∗ a set Z ⊆ G, which is a union of cosets of H with µ(Z) < ǫ,



µ((A\Z)△D) = 0.

(ii) (regularity) For any g ∈ G\Z, either µ(gH ∩ A) = 0 or µ(gH\A) = 0.

Proof. By Theorem 2.14(c) and Lemma 2.9(a, c), there is a decreasing sequence(Hi)

∞i=0 of θr-definable finite-index normal subgroups ofG such thatG00

θr =⋂∞

i=0Hi.By Lemma 2.16, there is a θr-definable set Z ′ ⊆ G and some i ∈ N such thatµ(Z ′) < ǫ and, if H := Hi, then for any g ∈ G\Z ′, either µ(gH ∩ A) = 0 orµ(gH\A) = 0. Let Z = {g ∈ G : gH ⊆ Z ′}. Then Z is a union of cosets of H ,and thus is θr-definable since [G : H ] is finite. Note that µ(Z) < ǫ since Z ⊆ Z ′.Moreover, if g ∈ G\Z then gH = g′H for some g′ ∈ G\Z ′, and so we have condition(ii). Now let D =

⋃{gH : g ∈ G\Z and µ(gH ∩ A) > 0}. Then D is θr-definable

since [G : H ] is finite. Moreover, µ((A\Z)△D) = 0 by condition (ii) and since[G : H ] is finite. So we have condition (i). �

We now prove structure and regularity for NIP sets in finite groups of uniformlybounded exponent. In this case, we obtain the optimal situation where NIP sets areentirely controlled by finite-index subgroups up to small error. This is related to asimilar result of Alon, Fox, and Zhao [3, Theorem 1.1] on finite abelian groups ofbounded exponent. Our result is stronger in the sense that the abelian assumptionis removed and the structural conclusions are improved, but also weaker in thesense that we do not obtain explicit bounds. This is analogous to the comparisonof our stable arithmetic regularity lemma in [14] (Theorem 1.1 above) to the workof the third author and Wolf [49] on stable sets in Fn

p .

Theorem 3.2. For any k, r ≥ 1 and ǫ > 0, there is n = n(k, r, ǫ) such that thefollowing holds. Suppose G is a finite group of exponent r and A ⊆ G is k-NIP.Then there are

∗ a normal subgroup H ≤ G of index at most n, and∗ a set Z ⊆ G, which is a union of cosets of H with |Z| < ǫ|G|,



|(A\Z)△D| < ǫ|H |.

(ii) (regularity) For any g ∈ G\Z, either |gH ∩A| < ǫ|H | or |gH\A| < ǫ|H |.


Proof. Note that condition (ii) follows immediately from condition (i). So supposecondition (i) is false. Then we have some fixed k, r ≥ 1 and ǫ > 0 such that, for


all i ∈ N, there is a finite group Gi of exponent r, which is a counterexample.Specifically, there is a k-NIP subset Ai ⊆ Gi such that, if H ≤ Gi is normal withindex at most i, and D,Z ⊆ Gi are unions of cosets of H with |Z| < ǫ|Gi|, then|(Ai\Z)△D| > ǫ|H |.

Let L be the group language with a new predicate A, and consider (Gi, Ai) asa finite L-structure. Let U be a nonprincipal ultrafilter on Z+, and let G be asufficiently saturated elementary extension of M :=

∏U (Gi, Ai). Let θ(x; y) be

the formula A(y · x). Note that θ(x; y) is invariant. Moreover, by Los’s Theoremand since M ≺ G, θ(x; y) is k-NIP (in G) and G has exponent r. So G/G00

θr is acompact torsion group, and thus is profinite by Fact 2.3(d). By Theorem 3.1, thereis a θr-definable finite-index normal subgroup H ≤ G and sets D,Z ⊆ G, whichare unions of cosets of H , such that µ(Z) < ǫ and µ((A\Z)△D) = 0.

Let n = [G : H ], and fix 〈θr〉-formulas φ(x; y), ψ(x; z), and ζ(x; u) such thatH , D, and Z are defined by instances of φ(x; y), ψ(x; z), and ζ(x; u), respectively.Given i ∈ N, let µi be the normalized counting measure on Gi. Let I be the set ofi ∈ N such that, for some tuples ai, bi, and ci from Gi,

(i) φ(x; ai) defines a normal subgroup Hi of Gi of index n,(ii) ψ(x; bi) and ζ(x; ci) define sets Di, Zi ⊆ Gi, respectively, which are each

unions of cosets of Hi, and(iii) µi(Zi) < ǫ and µi((Ai\Zi)△Di) <

ǫn .

Then I ∈ U by Los’s Theorem and since M ≺ G. So there is some i ∈ I such thati ≥ n, which contradicts the choice of (Gi, Ai). �

The statement of the previous result is almost identical to our result from [14] onstable arithmetic regularity in arbitrary finite groups (Theorem 1.1 above), exceptfor the presence of the error set Z. As in [14, Corollary 3.5], we can use this resultto deduce a very strong graph regularity statement for bipartite graphs defined byNIP subsets of finite groups of uniformly bounded exponent.

Let Γ = (V,W ;E) be a finite bipartite graph and fix nonempty sets X ⊆ V andY ⊆W . The edge density of the pair (X,Y ) is δΓ(X,Y ) := |(X×Y )∩E|/|X×Y |.We say that (X,Y ) is ǫ-regular if |δΓ(X,Y )− δΓ(X0, Y0)| ≤ ǫ for any X0 ⊆ X andY0 ⊆ Y such that |X0| ≥ ǫ|X | and |Y0| ≥ ǫ|Y |. Given vertices v ∈ V and w ∈ W ,define degΓ(v, Y ) = |{y ∈ Y : E(v, y)}| and degΓ(X,w) = |{x ∈ X : E(x,w)}|.Following [14], we say that the pair (X,Y ) is uniformly ǫ-good for Γ, whereǫ > 0, if |X | = |Y | and either:

(i) for any x ∈ X and y ∈ Y , degΓ(x, Y ) = degΓ(X, y) ≤ ǫ|X |, or(ii) for any x ∈ X and y ∈ Y , degΓ(x, Y ) = degΓ(X, y) ≥ (1 − ǫ)|X |.

One can show that if (X,Y ) is uniformly ǫ2-good then it is ǫ-regular, and eitherδΓ(X,Y ) ≤ ǫ or δΓ(X,Y ) ≥ 1−ǫ. In fact, a stronger property holds: if X0 ⊆ X andY0 ⊆ Y are nonempty and either |X0| ≥ ǫ|X | or |Y0| ≥ ǫ|Y |, then δΓ(X0, Y0) ≤ ǫor δΓ(X0, Y0) ≥ 1− ǫ (see [14, Proposition 3.4]).

Now suppose G is a finite group and A is a subset of G. Given X ⊆ G andg ∈ G, note that degΓG(A)(g,X) = |A ∩ gX | and degΓG(A)(X, g) = |A ∩ Xg|. Wenow observe that Theorem 3.2 implies a graph regularity statement for NIP subsetsof finite groups of uniformly bounded exponent, in which the partition is given bycosets of a normal subgroup and almost all pairs are uniformly good (and thusregular up to a change in ǫ). Given a group G, a normal subgroup H ≤ G, andC,D ∈ G/H , let C ·D denote the product of C and D in the quotient group G/H .


Corollary 3.3. For any k, r ≥ 1 and ǫ > 0 there is n(k, r, ǫ) such that the followingholds. Suppose G is a finite group of exponent r and A ⊆ G is k-NIP. Then there isa normal subgroup H of index n ≤ n(k, r, ǫ), and set Σ ⊆ (G/H)2, with |Σ| ≤ ǫn2,such that any (C,D) 6∈ Σ is uniformly ǫ-good for ΓG(A).

Proof. Fix k, r ≥ 1 and ǫ > 0 and let n(k, r, ǫ) be as in Theorem 3.2. Fix a finitegroup G and k-NIP set A ⊆ G. Then there is a normal subgroup H ≤ G of indexn ≤ n(k, r, ǫ), and a set I ⊆ G/H with |I| ≤ ǫn, such that for any C 6∈ I, either|C ∩ A| ≤ ǫ|H | or |C ∩ A| ≥ (1 − ǫ)|H |. Let Σ = {(C,D) ∈ (G/H)2 : C ·D 6∈ I}.Then Σ =

⋃C∈G/H{(C,C

-1 ·D) : D ∈ I}, and so |Σ| ≤ ǫn2. Finally, if (C,D) 6∈ Σ,

then (C,D) is uniformly ǫ-good for ΓG(A) (this is identical to the calculation inthe proof of [14, Corollary 3.5], and makes crucial use of normality of H). �

Remark 3.4. In Theorem 3.2, the assumption of uniformly bounded exponentwas used to obtain a certain profinite quotient, and so it is worth reviewing thisargument from a more general perspective. Specifically, fix k ≥ 1 and consider thefollowing property of a class G of finite groups: (∗)k For any sequences (Gi)

∞i=0 and

(Ai)∞i=0, where Gi ∈ G and Ai ⊆ Gi is k-NIP, and for any ultrafilter U on N, if G is a

sufficiently saturated elementary extension of∏

U(Gi, Ai), then G/G00θr is profinite,

where θ(x; y) := A(y · x). Then, for any G satisfying (∗)k and any ǫ > 0, there issome n = n(k, ǫ,G) such that any group G ∈ G and k-NIP set A ⊆ G satisfy theconclusions of Theorem 3.2 using n. Indeed, Theorem 3.2 only uses that for anyk, r ≥ 1, the class Gr of finite groups of exponent r satisfies (∗)k, for the ratherheavy-handed reason that compact torsion groups are profinite.

Profinite quotients also arise in the stable setting (recall that A ⊆ G is k-stableif ΓG(A) omits ([k], [k];≤) as an induced subgraph). In fact, if G is pseudofinite andsaturated, and θ(x; y) is a stable invariant formula, then the group G/G00

θr is finite(see [13, Corollary 3.17]). Therefore, in this case, the set E in Theorem 2.14(d)is empty since it is a Haar null set in a finite group. So, if one replaces “NIP”with “stable” in Lemma 2.16 and Theorem 3.1, then one can choose the error setZ = ∅, and a similar ultraproduct argument as in Theorem 3.2 yields Theorem 1.1(structure and regularity for stable subsets of finite groups).15

4. Bohr neighborhoods

In this section, we recall some basic definitions and facts concerning Bohr neigh-borhoods, and define an approximate version of Bohr neighborhoods, which we willneed for later arguments involving ultraproducts.

Given a group G, 1G denotes the identity (if G is abelian we use 0G). A Bohrneighborhood in a group G is a subset of the form π-1(U), where π : G→ L is ahomomorphism to a compact group L, with dense image, and U ⊆ L is an identityneighborhood (see, e.g., [6]). Under this definition, {1G} is a Bohr neighborhoodin any finite group G and so, in the setting of finite groups, one works with amore quantitative formulation (defined below). For our purposes, it will suffice toconsider the case that L is compact and metrizable. In particular, we will say thatthe pair (L, d) is a compact metric group if L is a compact metrizable groupand d is a bi-invariant metric on L compatible with the topology. By Fact 2.3(c), if

15It is worth noting that this explanation of Theorem 1.1 is not a faster proof than what isdone in [14]. In particular, [13, Corollary 3.17] relies on the same results from [27] used in [14] todirectly prove Theorem 1.1. Also, pseudofiniteness is not needed to prove that G/G00

θris finite.


L is any compact second-countable group then there is a (not necessarily unique)bi-invariant metric d on L such that (L, d) is a compact metric group.

Definition 4.1. Let H be a group and let (L, d) be a compact metric group. Givensome δ > 0 and a homomorphism τ : H → L, define

BLτ,δ = {x ∈ H : d(τ(x), 1L) < δ}.

A set B ⊆ H is a (δ, L)-Bohr neighborhood in H if B = BLτ,δ for some homo-

morphism τ : H → L.

Our ultimate goal is to transfer Bohr neighborhoods in pseudofinite groups toBohr neighborhoods in finite groups. To do this, we will need to approximate Bohrneighborhoods by definable objects. This necessitates an approximate notion of aBohr neighborhood, which involves approximate homomorphisms of groups.

Definition 4.2. Let H be a group and let (L, d) be a compact metric group.

(1) Given δ > 0, a function f : H → L is a δ-homomorphism if f(1H) = 1Land, for all x, y ∈ H , d(f(xy), f(x)f(y)) < δ.

(2) Given ǫ, δ > 0, a set Y ⊆ H is a δ-approximate (ǫ, L)-Bohr neighbor-hood in H if Y = {x ∈ H : d(f(x), 1L) < ǫ} for some δ-homomorphismf : H → L.

Approximate homomorphisms have been studied extensively in the literature,with a special focus on the question of when an approximate homomorphism is“close” to an actual homomorphism. For our purposes, this is what is needed toreplace approximate Bohr neighborhoods with actual Bohr neighborhoods. Moreprecisely, we will start with a definable approximate Bohr neighborhood in a pseu-dofinite group, and transfer this to find an approximate Bohr neighborhood in afinite group. At this point, we will be working with an approximate homomorphismfrom a finite group to a compact Lie group, which is a setting where one can alwaysfind a genuine Bohr neighborhood inside an approximate Bohr neighborhood, witha negligible loss in size.

Theorem 4.3 (Alekseev, Glebskiı, & Gordon [1, Theorem 5.13]). Let (L, d) be acompact metric Lie group. Then there is an αL > 0 such that, for any 0 < δ < αL,if H is a compact group and f : H → L is a δ-homomorphism, then there is ahomomorphism τ : H → L such that d(f(x), τ(x)) < 2δ for all x ∈ H.

An easy consequence is that in the setting of compact Lie groups, Bohr neigh-borhoods can be found inside approximate Bohr neighborhoods. Given a compactmetric group (L, d) and n ≥ 1, note that we have a compact metric group (Ln, dn),where Ln is endowed the product topology and dn(x, y) = max1≤i≤n d(xi, yi).

Corollary 4.4. Let (L, d) be a compact metric Lie group. Then there is an αL > 0such that, if H is a compact group, n ∈ N, and 0 < δ < αL, then any δ-approximate(3δ, Ln)-Bohr neighborhood in H contains a (δ, Ln)-Bohr neighborhood in H.

Proof. Fix αL > 0 from Theorem 4.3. Suppose H is a compact group, and Y ⊆ His a δ-approximate (3δ, Ln)-Bohr neighborhood in H , for some n ∈ N and 0 <δ < αL, witnessed by a δ-homomorphism f : H → Ln. We may assume n ≥ 1.For 1 ≤ i ≤ n, let fi : H → L be given by fi(x) = f(x)i. Then each fi isa δ-homomorphism. Given 1 ≤ i ≤ n, Theorem 4.3 provides a homomorphismτi : H → L such that d(fi(x), τi(x)) < 2δ for all x ∈ H . Let τ : H → Ln be such


that τ(x) = (τ1(x), . . . , τn(x)). Then τ is a homomorphism and dn(f(x), τ(x)) < 2δfor all x ∈ H . Now we have BLn

τ,δ ⊆ Y by the triangle inequality. �

The next result provides a lower bound on the size of Bohr neighborhoods infinite groups. The proof is a standard averaging argument (adapted from the abeliancase; see [48, Lemma 4.20] and/or [22, Lemma 4.1]). We include the details for thesake of clarity and to observe that the method works for Bohr neighborhoods innonabelian finite groups defined using compact metric groups.

Proposition 4.5. Let (L, d) be a compact metric group and, given δ > 0, letℓδ = ηL({t ∈ L : d(t, 1L) < δ}). For any finite group H and δ > 0, if B ⊆ H is a(2δ, L)-Bohr neighborhood in H, then |B| ≥ ℓδ|H |.

Proof. Fix a finite group H , a homomorphism τ : H → L, and some δ > 0. Givenx ∈ H , let fx : L→ {0, 1} be the characteristic function of {t ∈ L : d(τ(x), t) < δ}.Then

ℓδ|H | =∑

x∈H

∫

L

fx dηL =

∫

L

∑

x∈H

fx dηL.

So there must be some t ∈ L such that∑

x∈H fx(t) ≥ ℓδ|H |. In other words, ifS = {x ∈ H : d(τ(x), t) < δ} then |S| ≥ ℓδ|H |. Fix a ∈ S. For any x ∈ S, we have

d(τ(xa-1), 1L) = d(τ(x), τ(a)) ≤ d(τ(x), t) + d(τ(a), t) < 2δ.

Therefore Sa-1 ⊆ BLτ,2δ, and so |BL

τ,2δ| ≥ |Sa-1| = |S| ≥ ℓδ|H |. �

Remark 4.6. As we have seen in previous results (e.g., Lemma 2.9), real tori of theform Tr have a distinguished role in the study of compact abelian groups. Moreover,in the setting of finite abelian groups, Bohr neighborhoods are usually defined usinghomomorphisms to the torus (see, e.g., [7], [22]). Thus, in order to match thesedefinitions more explicitly, we define the metric dT1(x, y) = min{|x− y|, 1−|x− y|}on T1 (identified with [0, 1)) and the product metric dTr := dr

T1 on Tr for r ∈ N.Throughout the rest of the paper, when we speak of Tr as a compact metric group,will always work with this choice of metric.

Definition 4.7. Given a finite group H , a homomorphism τ : H → Tr, and someδ > 0, we let Br

τ,δ denote BTr

τ,δ. We call B ⊆ H a (δ, r)-Bohr neighborhood in

H if B = Brτ,δ for some τ .16

In Theorem 3.2, we showed that NIP sets in finite groups of uniformly boundedexponent are approximated by normal subgroups of uniformly bounded index. Asnoted in the introduction, we cannot expect such a result for NIP sets in finitegroups G of unrestricted exponent. So instead of subgroups of G, we will considerpairs (B,H), where H is a normal subgroup of G and B is a (δ, r)-Bohr neighbor-hood in H . Thus, in Proposition 4.9 below, we point out some ways in which Bohrneighborhoods behave like normal subgroups of “small” index. We will need thefollowing minor generalization of a well-known exercise, namely, if G is an amenablegroup and A ⊆ G has positive upper density then AA-1 is generic.

Proposition 4.8. Suppose G is a group, B is a left-invariant Boolean algebra ofsubsets of G, and ν is a left-invariant finitely additive probability measure on B.Suppose A ∈ B is such that ν(A) > 0. Then, for any Z ⊆ G, there is a finite setF ⊆ G\Z such that |F | ≤ 1

ν(A) and G\Z ⊆ FAA-1.

16In this case, r is sometimes referred to as the rank of B, and δ as the width of B.


Proof. We say that X ⊆ G separates A if xA∩ yA = ∅ for all distinct x, y ∈ X . Bythe assumptions on ν, if X ⊆ G separates A then |X | ≤ 1

ν(A) . Choose a finite set

F ⊆ G\Z with maximal size among subsets of G\Z that separate A. Fix x ∈ G\Z.Then there is y ∈ F such that xA ∩ yA 6= ∅, and so we may fix z ∈ xA ∩ yA. Theny-1z ∈ A and z-1x ∈ A-1, which means y-1x ∈ AA-1, and so x ∈ FAA-1. �

Proposition 4.9. Let G be a finite group and H ≤ G be a normal subgroup ofindex n. Suppose B is a (δ, r)-Bohr neighborhood in H, for some δ > 0 and r ∈ N.

(a) B = B-1, 1G ∈ B, and gB = Bg for any g ∈ G.(b) G is covered by at most n(2δ )r translates of B.

Proof. Part (a). We have B = B-1 by bi-invariance of dTr , and clearly 1G ∈ B.If g ∈ G and x ∈ B then gxg-1 ∈ H (since H is normal), and so gxg-1 ∈ B bybi-invariance of dTr . It follows that gB = Bg for any g ∈ G.

Part (b). For later reference, we prove something stronger.

Claim: For any Z ⊆ G there is F ⊆ G\Z such that |F | ≤ n(2δ )r and G\Z ⊆ FB.Proof: Suppose B = Br

τ,δ for some τ : H → Tr, and let B0 = Brτ,δ/2. Note that

ηTr({t ∈ Tr : dTr (t, 0Tr) < δ4}) = ( δ2 )r, and so |B0| ≥ ( δ2 )r|H | = n-1( δ2 )r|G| by

Proposition 4.5. By Proposition 4.8 (with ν the normalized counting measure onG), we have G\Z ⊆ FB0B

-10 for some F ⊆ G\Z with |F | ≤ n(2δ )r. Finally, note

that if x, y ∈ B0 then

d(τ(xy-1), 0Tr) = d(τ(x), τ(y)) ≤ d(τ(x), 0Tr ) + d(τ(y), 0Tr ) < δ.

So B0B-10 ⊆ B, and we have G\Z ⊆ FB. �

Note that part (a) of the previous fact holds for any compact metric group (L, d)in place of Tr (and does not use that G is finite); and the analogue of part (b) holdsfor any (L, d) with ( δ2 )r replaced by ℓδ/4 (from Proposition 4.5). We refer thereader to [48, Section 4.4] and [22, Sections 3 & 4] for more on the role of Bohrneighborhoods in arithmetic combinatorics and discrete Fourier analysis.

5. Structure and regularity: the general case

The next goal is a result analogous to Theorem 3.1, but without the assumptionthat G/G00

θr is profinite. For this, we need to understand more about descriptions ofG00

θr as an intersection of definable subsets of G. The goal is to find properties of de-finable sets which are both interesting algebraically, and also sufficiently first-orderso that they can be transferred to finite groups in arguments with ultraproducts.In particular, we will use approximate Bohr neighborhoods.

Suppose G is a group definable in a sufficiently saturated structure, and Γ ≤ Gis a type-definable normal subgroup of bounded index. By Lemma 2.9, Γ is anintersection of a bounded number of definable finite-index normal subgroups of Gwhenever G/Γ is profinite. The next result shows that in general, we can writeΓ =

⋂i∈I Wi where each Wi is a definable subset of a definable finite-index normal

subgroup Hi ≤ G, and there is a Bohr neighborhood Bi in Hi such that Γ ⊆Bi ⊆ Wi. Moreover, Bi is obtained from a definable homomorphism to a compactconnected Lie group (so, in particular, Bi is co-type-definable).

Proposition 5.1. Let G be a group definable in a sufficiently saturated structure.Suppose Γ ≤ G is type-definable and normal of bounded index. Then there is a


bounded jsl I and a decreasing net (Wi)i∈I of definable subsets of G such thatΓ =

⋂i∈I Wi and, for all i ∈ I, there are

∗ a definable finite-index normal subgroup Hi ≤ G,∗ a definable homomorphism πi : Hi → Li, where (Li, di) is a compact connectedmetric Lie group, and∗ a real number δi > 0,

such that Γ ⊆ BLi

πi,δi⊆Wi ⊆ Hi. Moreover:

(a) If Γ is countably-definable then we may assume I = N.(b) If G is an expansion of a group and Γ is θ-type-definable for some invariant

formula θ(x; y), then we may assume Wi, Hi, and πi are θ-definable.(c) If G is a pseudofinite expansion of a group then we may assume Li = Tni for

some ni ∈ N.(d) If G/Γ is abelian then we may assume Hi = G and Li = Tni for some ni ∈ N.

Proof. Let (Γi)i∈I0 and (Hi)i∈I0 be as in Lemma 2.9, where I0 is a small jsl. Foreach i ∈ I0, let Li = Hi/Γi and equip Li with some bi-invariant metric di (seeRemark 4.6). Let πi : Hi → Li be the canonical homomorphism. Then πi isdefinable since G→ G/Γi is definable.

For each i ∈ I0, let (W in)∞n=0 be a decreasing sequence of definable subsets of

G such that Γi =⋂∞

n=0Win. Let I be the set of all finite subsets of I0 × N, and

view I as a jsl under the subset ordering. Given σ = {(i1, n1), . . . , (ik, nk)} ∈ I,

let iσ = sup{i1, . . . , ik}, and set Hσ := Hiσ , Wσ := Hσ ∩⋂k

t=1Witnt

, Γσ := Γiσ ,Lσ = Liσ , and πσ = πiσ . Note that Γσ ⊆Wσ ⊆ Hσ for all σ ∈ I, and

⋂σ∈I Wσ = Γ.

By choice of iσ, we also have that (Wσ)σ∈I is decreasing.Now, given σ ∈ I, the set Uσ = {aΓσ ∈ G/Γσ : aΓσ ⊆ Wσ} is an identity

neighborhood in Lσ (by Fact 2.4(b)) and, by construction, π-1σ (Uσ) ⊆ Wσ . So

choose δσ > 0 such that Uσ contains the open ball of radius δσ around 1Lσ. This

finishes the proof of the main statement.We now deal with the remaining claims. Claims (b) and (c) follow by applying

Lemma 2.9(b, d) in the above construction. For claim (a), suppose Γ is countably-definable. Then, by Lemma 2.9(a), we may assume I0 = N. So I is the jsl offinite subsets of N × N under the subset ordering, which contains a cofinal sub-jslisomorphic to N.

Finally, for claim (d), suppose G/Γ is abelian. Then G/Γi is a compact abelianLie group for any i ∈ I0, and thus isomorphically embeds in Tni for some ni ∈ N

by Fact 2.3(f). So in the argument above, we can replace each Hi and πi with Gand G→ G/Γi ⊆ Tni , respectively. �

Note that in the previous result, the homomorphism πi : Hi → Li is also surjec-tive (except in part (d) where we can replace Hi with G when G/Γ is abelian).

One drawback of Proposition 5.1 is that the Bohr neighborhood BLi

πi,δiis not

necessarily definable. In order to work with definable objects, we will have toconsider approximate Bohr neighborhoods.

Definition 5.2. Let G be a group definable in a sufficiently saturated structure.Suppose H ≤ G is definable and π : H → L is a definable homomorphism to acompact metric group (L, d). Given t ≥ 1, we say that a sequence (Ym)∞m=0 ofsubsets of H is a definable (t, π)-approximate Bohr chain in H if (Ym)∞m=0 isdecreasing, kerπ =

⋂∞i=0 Ym, and there are (δm)∞m=0 and (fm)∞m=0 such that:


(i) (δm)∞m=0 is a decreasing sequence positive real numbers converging to 0,(ii) for all m, fm : H → L is a definable δm-homomorphism with finite image, and

(iii) for all m, Ym = {x ∈ H : d(fm(x), 1L) < tδm}.

Moreover, if G is an expansion of a group and H , π, and fm are all θ-definable, forsome fixed formula θ(x; y), then we say “θ-definable” in place of “definable”.

Proposition 5.3. Suppose (Ym)∞m=0 is a definable (respectively, θ-definable) (t, π)-approximate Bohr chain in H ≤ G, as in Definition 5.2. Then each set Ym is aδm-approximate (tδm, L)-Bohr neighborhood in H, and a definable (respectively,θ-definable) subset of G.17

Proof. The first claim is obvious. For definability, note that for any m ∈ N, H ispartitioned into finitely many fibers f -1

m(λ) for λ ∈ fm(H), each of which is definable(respectively, θ-definable) by Remark 2.6. Now Ym is a union of the (finitely many)fibers f -1

m(λ) where λ ∈ f(H) is such that d(λ, 1L) < ǫ. �

The parameter t in Definition 5.2 is introduced in order to control the “width”ǫ and the “error” δ in a δ-approximate (ǫ, L)-Bohr neighborhood. Specifically,it is desirable to have ǫ be some constant multiple t of δ, and in the followingresults we will choose t arbitrarily. This will eventually be used to find actual Bohrneighborhoods inside approximate Bohr neighborhoods, with L = Tn for somen ∈ N, in which case setting t = 3 will suffice (via Corollary 4.4).

Lemma 5.4. Let G be a group definable in a sufficiently saturated structure. Sup-pose H is a definable subgroup of G and π : H → L is a definable homomorphismto a compact metric group (L, d). Then, for any t ≥ 1, there is a definable (t, π)-approximate Bohr chain (Ym)∞m=0 in H. Moreover, if G is an expansion of a group,and H and π are θ-definable for some formula θ(x; y), then (Ym)∞m=0 is a θ-definable(t, π)-approximate Bohr chain.

Proof. Given λ ∈ L and ǫ > 0, let K(λ, ǫ) ⊆ L and U(λ, ǫ) ⊆ L be the closed ballof radius ǫ around λ and the open ball of radius ǫ around λ, respectively.

Fix m ≥ 1. Choose a finite set Λ ⊆ L such that L =⋃

λ∈ΛK(λ, 12m ) and

1L ∈ Λ. For any λ ∈ Λ, since π is definable, there is a definable set Dλ ⊆ Hsuch that π-1(K(λ, 1

2m )) ⊆ Dλ ⊆ π-1(U(λ, 1m )) (see Remark 2.6). Enumerate Λ =

{λ1, . . . , λk}, with λ1 = 1L, and, for 1 ≤ i ≤ k, let Di = Dλi. For 1 ≤ i ≤ k, define

Ei = Di\⋃

j<iDj . Then E1, . . . , Ek are definable and partitionH . This determines

a definable function fm : H → Λ such that fm(x) = λi if and only if x ∈ Ei. Forany x ∈ H , we have x ∈ Dfm(x) ⊆ π-1(U(fm(x), 1

m )), and so d(π(x), fm(x)) < 1m .

Note that fm(1H) = 1L by definition. Also, given x, y ∈ H , we have

d(fm(xy), fm(x)fm(y)) ≤ d(fm(xy), π(xy)) + d(π(x)π(y), fm(x)π(y))

+ d(fm(x)π(y), fm(x)fm(y))

= d(fm(xy), π(xy)) + d(π(x), fm(x)) + d(π(y), fm(y)) < 3m .

Altogether, fm : H → L is a 3m -homomorphism.

Now fix an integer t ≥ 1. For m ∈ N, define

Ym ={x ∈ H : d(fm(x), 1L) < 3t

m

}.

17On the other hand, we are not claiming that the family {Ym : m ∈ N} is uniformly definable(which could be misconstrued from our choice of terminology).


Note that D1 ⊆ Ym, and so kerπ ⊆ π-1(K(1L,12m )) ⊆ Ym. We now have a sequence

(Ym)∞m=1 of definable subsets of H , with kerπ ⊆ Ym for all m ∈ N. Moreover, forany m ∈ N, if x ∈ Ym then

d(π(x), 1L) ≤ d(π(x), fm(x)) + d(fm(x), 1L) < 3t+1m .

This implies kerπ =⋂∞

m=0 Ym. Finally, given m ∈ N, we have π-1(K(1L,12m )) ⊆

Ym ⊆ π-1(U(1L,3t+1m )). In particular, if n ≥ (6t + 2)m, then Yn ⊆ Ym. So, after

thinning the sequence, we may assume Ym+1 ⊆ Ym for all m ∈ N. Altogether, ifδm = 3

m , then (δm)∞m=1 and (fm)∞m=1 witness that (Ym)∞m=1 is a definable (t, π)-approximate Bohr chain in H .

For the “moreover” statement, suppose G is an expansion of a group, and Hand π are θ-definable for some formula θ(x; y). Then one can choose each Dλ tobe θ-definable (by Remark 2.6). So each fm is θ-definable by construction, and(Ym)∞m=1 is a θ-definable (t, π)-approximate Bohr chain. �

We now combine the above ingredients to prove a structure and regularity the-orem for θr-definable sets in a sufficiently saturated pseudofinite group G, whereθ(x; y) is an arbitrary invariant NIP formula.

Theorem 5.5. Let G be a sufficiently saturated pseudofinite expansion of a group,and suppose θ(x; y) is an invariant NIP formula. Fix a θr-definable set A ⊆ G andsome ǫ > 0. Then there are

∗ a θr-definable finite-index normal subgroup H ≤ G,∗ a θr-definable homomorphism π : H → Tn, for some n ∈ N, and∗ a θr-definable set Z ⊆ G, with µ(Z) < ǫ,

such that, for any integer t ≥ 1, there is

∗ a θr-definable (t, π)-approximate Bohr chain (Ym)∞m=0 in H,

satisfying the following properties, for any m ∈ N.

(i) (structure) There is a set Dm ⊆ G, which is a finite union of left translatesof Ym, such that

µ((A△Dm)\Z) = 0.

(ii) (regularity) For any g ∈ G\Z, either µ(gYm ∩ A) = 0 or µ(gYm\A) = 0.

Moreover, if G/G00θr is abelian then we may assume H = G.

Proof. Let (Wi)∞i=0 be a sequence of θr-definable sets in G satisfying the conditions

of Proposition 5.1 with Γ = G00θr . By Lemma 2.16, there is a θr-definable set Z ⊆ G

and some i ∈ N such that µ(Z) < ǫ and, if W := Wi, then for all g ∈ G\Z, we haveµ(gW ∩ A) = 0 or µ(gW\A) = 0. Proposition 5.1 associates to W a θr-definablehomomorphism π : H → Tn, whereH is a θr-definable finite-index normal subgroupof G. If G/G00

θr is abelian then we may further assume H = G. Fix t ≥ 1. ByLemma 5.4, there is a θr-definable (t, π)-approximate Bohr chain (Ym)∞m=0 in H .Recall that each Ym is definable by Proposition 5.3. Since kerπ is type-definableand contained in the definable set W , it follows from saturation that Ym ⊆ W forsufficiently large m. So for sufficiently large m we have that, for any g ∈ G\Z,either µ(gYm ∩ A) = 0 or µ(gYm\A) = 0. Thus, after removing finitely many setsYm from the sequence, we have condition (ii).

Toward proving condition (i), fix m ∈ N. Since kerπ is a subgroup of H , we mayuse saturation (similar to as in the proof of Lemma 2.16), to find some r ≥ m suchthat YrY

-1r ⊆ Ym. Since Yr contains a type-definable bounded-index subgroup of


G (namely, kerπ), it follows that Yr is generic and so µ(Yr) > 0. By Proposition4.8, there is a finite set F ⊆ G\Z such that G\Z ⊆ FYm. Let I be the set of g ∈ Fsuch that µ(gYm\A) = 0, and note that if g ∈ F\I then µ(gYm ∩ A) = 0. LetDm = IYm. Since G\Z ⊆ FYm, we have

A△Dm ⊆ Z ∪⋃

g∈I

(gYm\A) ∪⋃

g∈F\I

(gYm ∩ A),

and so µ((A△Dm)\Z) = 0. �

The next goal is our main result for NIP sets in arbitrary finite groups (seeTheorem 5.7). Roughly speaking, we will show that if A is a k-NIP set in a finitegroup G, then there is a normal subgroup H ≤ G, and Bohr neighborhood B in H ,such that almost all translates of B are almost contained in A or almost disjointfrom A. Moreover, A is approximately a union of translates of B, and the index ofH and complexity of B are bounded in terms of k and ǫ.

The proof of Theorem 5.7 from Theorem 5.5 is of course in analogy to theproof of Theorem 3.2 from Theorem 3.1. However, the argument is significantlymore complicated, due to certain crucial differences between Bohr neighborhoodsand subgroups. Specifically, Bohr neighborhoods are not closed under the groupoperation and, moreover, distinct translates of a Bohr neighborhood need not bedisjoint.18 Thus, when using Theorem 5.5 to obtain results for finite groups, we willfocus solely on the regularity statement. In order to then deduce structure fromregularity in finite groups, we will need to argue similarly as in the end of the proofof Theorem 5.5, while also taking into account the quantitative behavior of Bohrneighborhoods described in Propositions 4.5 and 4.9. So we first prove a lemma,which gives a rather flexible version of the regularity statement, and also containsthe ultraproduct argument necessary to prove Theorem 5.7.

Lemma 5.6. For any k ≥ 1 and ǫ > 0, and any function γ : (Z+)2 × (0, 1]→ R+,there is n = n(k, ǫ, γ) such that the following holds. Suppose G is a finite groupand A ⊆ G is k-NIP. Then there are

∗ a normal subgroup H ≤ G of index m ≤ n,∗ a (δ, r)-Bohr neighborhood B in H and a δ-approximate (3δ, r)-Bohr neighbor-hood Y in H, where r ≤ n and 1

n ≤ δ ≤ 1, and∗ a set Z ⊆ G, with |Z| < ǫ|G|,

such that B ⊆ Y ⊆ H and, for any g ∈ G\Z, either |gY ∩ A| < γ(m, r, δ)|B|or |gY \A| < γ(m, r, δ)|B|. Moreover, H, Y , and Z are in the Boolean algebragenerated by {gAh : g, h ∈ G}, and if G is abelian then we may assume H = G.

Proof. Suppose not. Then we have k ≥ 1, ǫ > 0, and γ : (Z+)2 × (0, 1] → R+

witnessing this. In particular, for any n ≥ 1, there is a finite group Gn and a k-NIPsubset An ⊆ Gn such that, for any H,B, Y, Z ⊆ Gn, if

∗ H is a normal subgroup of Gn of index m ≤ n, and H = Gn if Gn is abelian,∗ H , Y , and Z are in the Boolean algebra generated by {gAnh : g, h ∈ Gn},∗ B is a (δ, r)-Bohr neighborhood in H and Y is a δ-approximate (3δ, r)-Bohr

neighborhood in H , where r ≤ n and 1n ≤ δ ≤ 1,

∗ |Z| < ǫ|Gn|, and B ⊆ Y ⊆ H ,

18This is also the reason why Theorem 5.7 does not yield a graph regularity statement forΓG(A) involving a partition into translates of Bohr neighborhoods.


then there is some g ∈ Gn\Z such that |gY ∩ An| ≥ γ(m, r, δ)|B| and |gY \An| ≥γ(m, r, δ)|B|.

Let L be the group language together with an extra predicate A, and considereach (Gn, An) as a finite L-structure. Let U be a nonprincipal ultrafilter on Z+,and let G be a sufficiently saturated elementary extension of M :=

∏U (Gn, An).

We identify A = A(G). Let θ(x; y) be the formula A(y · x). Note that θ(x; y)is invariant, and k-NIP (in G) by Los’s Theorem and since M ≺ G. Finally, letα := αT1 > 0 be as in Corollary 4.4. By Theorem 5.5 (with t = 3), there areθr-definable sets Y, Z ⊆ G, a θr-definable finite-index normal subgroup H ≤ G,and a θr-definable δ-homomorphism f : H → Tr, for some r ∈ N and 0 < δ < α,such that:

∗ if G is abelian then H = G,∗ µ(Z) < ǫ,∗ f(H) is finite and Y = {x ∈ H : dTr (f(x), 0Tr) < 3δ}, and∗ for any g ∈ G\Z, either µ(gY ∩ A) = 0 or µ(gY \A) = 0.

Let m = [G : H ], and set ǫ∗ = γ(m, r, δ)m-1δr > 0. Let Λ = f(H) and, givenλ ∈ Λ, let F (λ) = f -1(λ). Then each F (λ) is θr-definable by Remark 2.6.

Fix 〈θr〉-formulas φ(x; y), ψ(x; z), ζ(x; u), and ξλ(x; vλ) for λ ∈ Λ (withoutparameters) such that H , Y , Z, and F (λ) for λ ∈ Λ are defined by instances ofφ(x; y), ψ(x; z), ζ(x; u), and ξλ(x; vλ), respectively. Given n ∈ N, let µn denote thenormalized counting measure on Gn. Let d denote dTr . Define I ⊆ N to be the setof n ∈ N such that, for some tuples an, bn, cn, and dn,λ for λ ∈ Λ,

(i) φ(x; an) defines a normal subgroup Hn of Gn of index m,(ii) ζ(x; cn) defines a subset Zn with µn(Zn) < ǫ,

(iii) for each λ ∈ Λ, ξλ(x; dn,λ) defines a subset Fn(λ) of Hn, and (Fn(λ))λ∈Λ

forms a partition of Hn,(iv) if fn : Hn → Λ is defined so that fn(x) = λ if and only if x ∈ Fn(λ), then fn

is a δ-homomorphism from Hn to Tr,(v) ψ(x; bn) defines the set Yn = {x ∈ Hn : d(fn(x), 0Tr ) < 3δ}, and

(vi) for all g ∈ Gn\Zn, either µn(gYn ∩ An) < ǫ∗ or µn(gYn\An) < ǫ∗.

We claim that I ∈ U by Los’s Theorem and since M ≺ G. In other words, theclaim is that conditions (i) through (vi) are first-order expressible (possibly usingthe expanded language discussed in Remark 2.13). This is clear for (i), (ii), (iii),and (vi), and the only subtleties lie in (iv) and (v). In both cases, the crucial pointis that Λ is finite, and so these conditions can be described by first-order sentences(using similar ideas as in Proposition 5.3). For instance, to express condition (iv),fix λ ∈ Λ, and let Pλ = {(λ1, λ2) ∈ Λ2 : d(λ, λ1 + λ2) < δ}. Let σλ be a sentenceexpressing that for any x, y, if x · y ∈ F (λ) then x ∈ F (λ1) and y ∈ F (λ2) forsome (λ1, λ2) ∈ P (λ). Then the conjunction

∧λ∈Λ σλ expresses precisely that f is

a δ-homomorphism. The details for condition (v) are similar and left to the reader.Since I ∈ U , we may fix n ∈ I such that n ≥ max{m, r, δ-1}. Since 0 <

δ < α and Yn is a δ-approximate (3δ, r)-Bohr neighborhood in Hn, it follows fromCorollary 4.4 that Yn contains a (δ, r)-Bohr neighborhood B in Hn. So, by choice of(Gn, An), there must be g ∈ Gn\Zn such that µn(gYn ∩An) ≥ γ(m, r, δ)µn(B) andµn(gYn\An) ≥ γ(m, r, δ)µn(B). So, to obtain a contradiction (to (vi)), it sufficesto show that ǫ∗ ≤ γ(m, r, δ)µn(B), i.e. (by choice of ǫ∗), show m-1δr ≤ µn(B).To see this, note that |B| ≥ δr|Hn| by Proposition 4.5 (applied with L = Tr, soℓδ/2 = δr), and so |B| ≥ δrm-1|Gn| since [Gn : Hn] = m. �


We now prove the main result for NIP subsets of arbitrary finite groups.

Theorem 5.7. For any k ≥ 1 and ǫ > 0 there is n = n(k, ǫ) such that the followingholds. Suppose G is a finite group and A ⊆ G is k-NIP. Then there are

∗ a normal subgroup H ≤ G of index m ≤ n,∗ a (δ, r)-Bohr neighborhood B in H, where r ≤ n and 1

n ≤ δ ≤ 1, and∗ a subset Z ⊆ G, with |Z| < ǫ|G|,


(i) (structure) There is a set D ⊆ G, which is a union of at most m(2δ )r translatesof B, such that

|(A△D)\Z| < ǫ|B|.

(ii) (regularity) For any g ∈ G\Z, either |gB ∩ A| < ǫ|B| or |gB\A| < ǫ|B|.

Moreover, H and Z are in the Boolean algebra generated by {gAh : g, h ∈ G}, andif G is abelian then we may assume H = G.

Proof. Fix k ≥ 1 and ǫ > 0. Define γ : (Z+)2 × (0, 1] → R+ such that γ(x, y, z) =ǫx-1( z2 )y. Let n = n(k, ǫ, γ) be given by Lemma 5.6. Fix a finite group G and ak-NIP subset A ⊆ G. By Lemma 5.6, there are

∗ a normal subgroup H ≤ G of index m ≤ n,∗ a subset Y ⊆ H ,∗ a (δ, r)-Bohr neighborhood B in H , where r ≤ n and 1

n ≤ δ ≤ 1, and∗ a set Z ⊆ G, with |Z| < ǫ|G|,

such that, B ⊆ Y ⊆ G and, for all g ∈ G\Z, either |gY ∩ A| < γ(m, r, δ)|B| or|gY \A| < γ(m, r, δ)|B|. Moreover, if G is abelian then we may assume H = G.Since B ⊆ Y and γ(m, r, δ) ≤ ǫ, this immediately yields condition (ii).

For condition (i), we argue as in the proof of Theorem 5.5. First, by the claim inthe proof of Proposition 4.9(b), there is a set F ⊆ G\Z such that |F | ≤ m(2δ )r andG\Z ⊆ FB. Let I = {g ∈ F : |gB\A| < γ(m, r, δ)|B|}, and note that if g ∈ F\Ithen |gB ∩ A| < γ(m, r, δ)|B|. Let D = IB. Since G\Z ⊆ FB, we have

A△D ⊆ Z ∪⋃

g∈I

(gB\A) ∪⋃

g∈F\I

(gB ∩ A), and so

|(A△D)\Z| ≤∑

g∈I

|gB\A|+∑

g∈F\I

|gB ∩ A| < |F |γ(m, r, δ)|B| ≤ ǫ|B|. �

We end this section with some remarks on NIP subsets of finite simple groups.To motivate this, we first consider the stable case. In particular, given k ≥ 1 andǫ > 0, it follows immediately from Theorem 1.1 that if G is a sufficiently large(depending on k and ǫ) finite simple group, and A ⊆ G is k-stable, then |A| < ǫ|G|or |A| > (1 − ǫ)|G|.19 On the other hand, this conclusion fails for NIP subsets ofabelian finite simple groups (e.g., by the examples of NIP sets in Z/pZ mentionedin the introduction). So it is interesting to observe that for nonabelian finite simplegroups, one recovers the same “triviality” for NIP sets.

Corollary 5.8. Given k ≥ 1 and ǫ > 0, there is m = m(k, ǫ) such that the followingholds. Suppose G is a nonabelian finite simple group, with |G| > m, and A ⊆ G isk-NIP. Then |A| < ǫ|G| or |A| > (1− ǫ)|G|.

19If G is simple and abelian (i.e., cyclic of prime order), then one could further use the proofof [50, Corollary 3] to obtain an explicit lower bound on how large G needs to be.


Proof. Let m = n(k, ǫ2 ) be as in Theorem 5.7. Suppose G is a nonabelian finite

simple group, with |G| > m, and A ⊆ G is k-NIP. Then we have a normal subgroupH ≤ G of index at most m, a (δ, r)-Bohr neighborhood B in H (for some δ andr), a set Z ⊆ G with |Z| < ǫ

2 |G|, and a union D of translates of B such that|(A△D)\Z| < ǫ

2 |B|. So |A△D| ≤ |(A△D)\Z| + |Z| < ǫ|G|. Since G is simpleand |G| > m, we have H = G. Now B contains the kernel K of a homomorphismfrom G to Tr. So G/K is abelian, which implies K = B = G since G is simple andnonabelian. Now D is either ∅ or G, and the result follows. �

The previous proof highlights the significance of obtaining Bohr neighborhoodsdefined from abelian compact Lie groups, which is ultimately possible thanks toTheorem 2.7. In [12], the first author used techniques of Alon, Fox, and Zhao [3]and a result of Gowers [20] on “quasirandom” groups to give a different proof of theprevious corollary, which yields m = exp(c(90/ǫ)6k−6) for some absolute constantc (see [12, Remark 8.7]).

6. Distal arithmetic regularity

In this section, we adapt the preceding results to the case of NIP fsg groups withsmooth left-invariant measures (e.g., those definable in distal theories). In contrastto previous results, where we focused on a single NIP formula, here we will operatein the setting of groups definable in an NIP theory.

In this section, we let T be a complete theory and we work in a saturated modelM∗. Given M �M∗ and an M -definable set X in M∗, a Keisler measure on Xover M is a finitely additive probability measure defined on the Boolean algebraof M -definable subsets of X . If M∗ = M then µ is a global Keisler measure onX . If T is NIP, then a global Keisler measure µ on a definable set X is genericallystable if there is a small model M ≺ M∗ such that X is M -definable, µ is M -definable (i.e., for any formula φ(x; y) with x in the same sort as X , the mapb 7→ µ(φ(x; b) ∩X) is M -definable) and µ is finitely satisfiable in M (i.e., if Y ⊆ Xis definable and µ(Y ) > 0 then Y ∩M 6= ∅). See [44, Section 7.5]) for details.

Definition 6.1. (T is NIP.) A definable group G is fsg if it admits a genericallystable left-invariant (global) Keisler measure.

We note that this is not the original definition of fsg (which is given in [26]),but rather the right characterization (in NIP theories) for our purposes (see [26,Proposition 6.2] and [29, Remark 4.2]). The significance of this notion in the contextof our work is illustrated by the following example.

Example 6.2. If T is NIP and G is a definable pseudofinite group, then G is fsgsince the pseudofinite counting measure on G is generically stable. This followsdirectly from the VC-Theorem (see, e.g., [44, Example 7.32], [13, Section 2]).

The fsg property for a definable group in an NIP theory has strong consequences.For example, such a group has a unique left-invariant Keisler measure, which is alsothe unique right-invariant Keisler measure [28, Theorem 7.7]. Moreover, genericityand positive measure coincide (as in Theorem 2.14(a); see the proof of [26, Proposi-tion 6.2]). Such groups also satisfy a generic compact domination statement similarto Theorem 2.14(d) (see [45, Corollary 4.9]). However, in this section, we focus ona certain strengthening of generic compact domination for definable fsg groups inNIP theories whose unique left-invariant Keisler measure is smooth.


Definition 6.3. Let X be definable in M∗. A global Keisler measure µ on X issmooth if there is a small model M ≺ M∗ such that X is M -definable and µ isthe unique global Keisler measure on X extending µ|M .

If T is NIP, then any smooth measure is generically stable (see Proposition 7.10and Theorem 7.29 in [44]), and so any definable group with a smooth left-invariantKeisler measure is fsg. A place to find smooth measures is in the setting of distaltheories, which were introduced by Simon in [43]. For our purposes we will takethe following characterization as a definition of distality (see [43, Theorem 1.1]).

Definition 6.4. T is distal if it is NIP and every global generically stable Keislermeasure is smooth.

So, in particular, if T is distal and G is a definable fsg group, then the unique left-invariant Keisler measure on G is smooth. In [44, Proposition 8.41], Simon showsthat if T is countable and NIP, and G is a definable group with a smooth left-invariant Keisler measure, then generic compact domination for G can be strength-ened to outright “compact domination”. We give a formulation of this result suit-able for our applications. First, recall that if T is NIP and G is a definable group,then G has a smallest type-definable subgroup of bounded index, denoted G00,which is an intersection of at most |T | definable sets (see [42] and/or [26, Proposi-tion 6.1]). The reader should compare the following result to Lemma 2.16.

Lemma 6.5. Assume T is countable and NIP, and G is a definable fsg group. Letµ be the unique left-invariant Keisler measure on G, and assume µ is smooth. Let(Wi)

∞i=0 be a decreasing sequence of definable sets such that G00 =

⋂∞i=0Wi. Fix a

definable set A ⊆ G. Then, for any ǫ > 0, there is a definable set Z ⊆ G and somei ∈ N such that µ(Z) < ǫ and, for any g ∈ G\Z, either gWi ∩ A = ∅ or gWi ⊆ A.

Proof. Let F = {C ∈ G/G00 : C ∩A 6= ∅ and C ∩ (G\A) 6= ∅}. By [44, Proposition8.41], F is closed and ηG/G00(F ) = 0. Now fix ǫ > 0. By Fact 2.4(e), there is

a definable set Z ⊆ G such that µ(Z) < ǫ and {a ∈ G : aG00 ∈ F} ⊆ Z. Bysaturation of G and since (Wi)

∞i=0 is decreasing, there is some i ∈ N such that for

any g ∈ G\Z, either gWi ∩ A = ∅ or gWi ⊆ A. �

We now prove analogues of Theorems 3.1 and 5.5 for an fsg group G, definable inan NIP theory, such that the unique left-invariant Keisler measure on G is smooth.In these results, the assumptions are stronger in the sense that the whole theory isassumed to be NIP. Moreover, in the conclusions we have definability of the data,but no claims about definability in a certain Boolean fragment (see Remark 6.8).On the other hand, we have outright inclusion or disjointness, rather than up to ǫ,which yields stronger structure and regularity statements.

Theorem 6.6. Assume T is NIP. Let G be a definable fsg group, and let µ bethe unique left-invariant Keisler measure on G. Suppose µ is smooth (e.g., if T isdistal) and G/G00 is profinite. Fix a definable set A ⊆ G and some ǫ > 0. Thenthere are

∗ a definable finite-index normal subgroup H of G, and∗ a set Z ⊆ G, which is a union of cosets of H with µ(Z) < ǫ,


(i) (structure) A\Z is a union of cosets of H.(ii) (regularity) For any g ∈ G\Z, either gH ∩ A = ∅ or gH ⊆ A.


Proof. The proof is similar to that of Theorem 3.1. First, we may restrict to acountable language in which G and A are definable and µ is still smooth (see [44,Lemma 7.8] and the remarks after [44, Proposition 8.38]).

By Lemma 2.9, we may fix a decreasing sequence (Hi)∞i=0 of definable finite-

index normal subgroups of G such that G00 =⋂∞

i=0Hi. By Lemma 6.5, we havea definable set Z and some H = Hi satisfying condition (ii). As in the proof ofTheorem 3.1, we may assume Z is a union of cosets of H . For condition (i), setD =

⋃{g ∈ G : gH ⊆ A and gH ∩ Z = ∅}. Then D is a union of cosets of H , and

D ⊆ A ⊆ D ∪Z. Since Z is a union of cosets of H , it follows from the definition ofD that D ∩ Z = ∅. So A\Z = D. �

Now we prove the general statement.

Theorem 6.7. Assume T is NIP. Let G be a definable fsg group, and let µ bethe unique left-invariant Keisler measure on G. Suppose µ is smooth (e.g., if T isdistal). Fix a definable set A ⊆ G and some ǫ > 0. Then there are

∗ a definable finite-index normal subgroup H of G,∗ a compact connected metric Lie group (L, d),∗ a definable homomorphism π : H → L, and∗ a definable set Z ⊆ G, with µ(Z) < ǫ,

such that, for any integer t ≥ 1, there is

∗ a definable (t, π)-approximate Bohr chain (Ym)∞m=0 in H,

satisfying the following properties, for any m ∈ N.

(i) (structure) There is Dm ⊆ G, which is a finite union of left translates of Ym,such that Dm ⊆ A ⊆ Dm ∪ Z.

(ii) (regularity) For any g ∈ G\Z, either gYm ∩ A = ∅ or gYm ⊆ A.

Moreover, if G is pseudofinite then we may assume L = Tn for some n ∈ N; and ifG/G00 is abelian then we may assume L = Tn for some n ∈ N and H = G.

Proof. The proof is similar to that of Theorem 5.5. First, as before, we may assumeT is in a countable language. Let (Wi)

∞i=0 be a decreasing sequence of definable

subsets of G satisfying the conditions of Proposition 5.1, with Γ = G00. By Lemma6.5, there is a definable set Z ⊆ G and some i ∈ N such that µ(Z) < ǫ and, ifW := Wi, then for all g ∈ G\Z, either gW ∩ A = ∅ or gW ⊆ A. Proposition 5.1associates to W a definable homomorphism π : H → L, whereH is a definable finite-index normal subgroup of G and (L, d) is compact connected metric Lie group. Notealso that the “moreover” statement is given by Proposition 5.1. By Lemma 5.4,there is a definable (t, π)-approximate Bohr chain (Ym)∞m=0 in H . By saturation, ifm is sufficiently large then Ym ⊆W and so if g ∈ G\Z then either gYm ∩A = ∅ orgYm ⊆ A. This yields condition (ii). For condition (i), mimic the end of the proofof Theorem 5.5 to find Dm ⊆ G, which is a finite union of left translates of Ym,such that Dm ⊆ X ⊆ Dm ∪ Z (replace each occurrence of µ(X ∩ gYm) = 0 withX ∩ gYm = ∅, and each occurrence of µ(X\gYm) = 0 with gYm ⊆ X). �

Since Theorems 6.6 and 6.7 use global assumptions on the theory, a fully generalstatement for applications to finite groups would be rather cumbersome to state.However, one concludes, in a routine fashion, structure and regularity for suitablefamilies of finite groups, for example a collection G of finite L-structures expandinggroups, such that any completion of Th(G) is distal. In the next section, we will


discuss an application to the family of finite groups obtained as quotients of acompact p-adic Lie group by its open normal subgroups.

Remark 6.8. Results along the lines of Theorems 3.1 and 5.5 can also be shownfor fsg groups definable in NIP theories, but without the smoothness assumption.Indeed, such groups satisfy generic compact domination just as in Theorem 2.14(d),but with G00

θr replaced by G00 (see [45, Corollary 4.9]). However, as discussed in[13, Remark 1.2], since the unique left-invariant measure on an NIP fsg group Gis generically stable, one could fix an invariant formula θ(x; y) and construct G00

θr

as in Theorem 2.14. This yields structure and regularity theorems as before withadditional information about the definability of the data. Precisely:

Assume T is NIP. Suppose G is definable and fsg, and let µ be the unique left-invariant Keisler measure on G. Fix an invariant formula θ(x; y), where x is inthe sort for G. Then, for any θr-definable A ⊆ G and any ǫ > 0, we have theconclusion of Theorem 5.5, except with Tn replaced by some compact connectedmetric Lie group (L, d). If G/G00

θr is profinite, then the conclusion of Theorem 3.1holds exactly as stated.

It would be interesting to pursue notions of smoothness for local measures, or“local distality” for formulas, and recover local versions of Theorems 6.7 and 6.6.For instance, one might consider an NIP formula θ(x, y) such that every genericallystable global Keisler measure on the Boolean algebra of θ-formulas is smooth.

7. Compact p-adic analytic groups

In this section, we apply Theorem 6.6 to the setting of compact p-adic analyticgroups (see Theorems 7.3 and 7.5). We assume some familiarity with the p-adicfield Qp and p-adic model theory. See [5] for further reading. The topology on Qp

is given by the valuation v where open neighborhoods of a point a are defined byv(x − a) ≥ n for n ∈ Z. The topology on Qn

p is the product topology. A p-adicanalytic function is a function f , from some open V ⊆ Qn

p to Qp, such that forevery a ∈ V , there is an open neighborhood of a ∈ V in which f is given by aconvergent power series. We obtain the notions of a p-adic analytic manifold anda p-adic analytic (or Lie) group.

We let Qanp denote the expansion of the field (Qp,+, ·) by symbols for all conver-

gent (in Zp) power series in Zp[[X1, . . . , Xn]] for all n. Then any compact p-adicanalytic manifold or group is seen to be naturally definable in the structure Qan

p (weconflate definable and interpretable at this point). It is well known that Th(Qan

p )is distal, and that distality passes to T eq (see Exercise 9.12 of [44]).

The following fact is pointed out in [33] (with origins in du Sautoy’s Ph.D. thesis).See [33, Proposition 1.2] and its proof.

Fact 7.1. Let K be a compact p-adic analytic group. Then the family of opennormal subgroups is uniformly definable in Qan

p .

In particular, the collection of quotients of a compact p-adic analytic group byits open normal subgroups is a family of uniformly definable (in Qan

p ) finite groups.Let us fix a compact p-adic analytic group K (so definable in Qan

p ). If M∗ ≻ Qanp

is sufficiently saturated, then K(M∗) denotes the group definable in the structureM∗ by the same formula as the one defining K in Qan

p . The next result combinesthe “Claim” in [26, Section 6] with Corollaries 2.3 and 2.4 of [38].


Theorem 7.2. [26, 38] Let M∗ ≻ Qanp be sufficiently saturated. Then K(M∗) is

fsg. Moreover, K(M∗)/K(M∗)00 is isomorphic to K, and thus is profinite. Inparticular, K(M∗)00 =

⋂∞i=0Hi(M

∗) where {Hi : i ∈ N} is the neighborhood basisat the identity consisting of open normal subgroups of K.

Our first application of Theorem 6.6 is the following “structure and regularity”statement for definable subsets of compact p-adic analytic groups. In fact, as wepoint out below, this can also be seen as a fairly direct application of [38, Proposition2.8], and an extension of certain results in that paper.

Theorem 7.3. Let K be a compact p-adic analytic group (so definable in thestructure Qan

p ). Let A ⊆ K be definable in Qanp , and let ǫ > 0. Then there are

∗ an open (so finite-index) normal subgroup H of K, and∗ a set Z ⊆ K, which is a union of cosets of H with ηK(Z) < ǫ,


(i) (structure) A\Z is a union of cosets of H.(ii) (regularity) For any g ∈ K\Z, either gH ∩A = ∅ or gH ⊆ A.

Proof. LetM∗ ≻ Qanp be sufficiently saturated. By Fact 7.2, we may apply Theorem

6.6 to the group K(M∗) and the definable set A(M∗), and further assume that thefinite-index normal subgroup obtained is H(M∗) for some open normal subgroupH of K. So the error set is of the form Z(M∗), where Z is definable in Qan

p .Note also that if µ is the (unique) left-invariant Keisler measure on K(M∗), thenµ(Z(M∗)) = ηK(Z) since H is a finite index subgroup of K and thus µ(H(M∗)) =ηK(H) = 1/[K : H ]. Altogether, by elementarity, we have (i) and (ii). �

Remark 7.4.

(1) The proof of Theorem 7.3 only requires definability of the open normalsubgroups of K (and not uniform definability).

(2) Proposition 2.8 of [38] states that K(M∗) is compactly dominated via themap K(M∗)→ K(M∗)/K(M∗)00. So we could also have deduced Theorem7.3 from this result, together with the standard methods.

(3) It would be interesting to prove Theorem 7.3 from the cell decompositionresults of Denef and others (at least when K = Zn

p for some n).

Our second application of Theorem 6.6 is to the family of quotients of a compactp-adic analytic group by its open normal subgroups.

Theorem 7.5. Let K be a compact p-adic analytic group. Let (Gi)i∈I be the familyof finite groups obtained as quotients of K by open normal subgroups. Let A ⊆ Kbe definable in Qan

p and for i ∈ I, let Ai ⊆ Gi be the image of A under the quotientmap. Fix ǫ > 0. There is some n = n(K,A, ǫ) such that for any i ∈ I, there are

∗ a normal subgroup Hi ≤ Gi of index at most n, and∗ a set Zi ⊆ Gi, which is a union of cosets of Hi with |Zi| < ǫ|Gi|,


(i) (structure) Ai\Zi is a union of cosets of Hi.(ii) (regularity) For any g ∈ Gi\Zi, either gHi ∩Ai = ∅ or gHi ⊆ Ai.

Proof. It suffices to prove condition (i). Toward a contradiction, suppose we haveǫ > 0 such that, for any n ≥ 1 there is some in ∈ I such that if H ≤ Gin is a


normal subgroup of index at most n, and Z ⊆ Gin is a union of cosets of H with|Z| < ǫ|Gin |, then Ain\Z is not a union of cosets of H .

Let Gi = K/Ui, where (Ui)i∈I lists the open normal subgroups of K. By Fact7.1, we may view Ut as a formula (over Qp) in the variable t, and I as a definableset in Qan

p in the sort for t. Let SI(Qp) be the space of types with parametersfrom Qp concentrating on I. Let M∗ ≻ Qan

p be sufficiently saturated, and leti∗ ∈ I(M∗) realize an accumulation point of {tp(in/Qp) : n ≥ 1} in SI(Qp). LetG = K(M∗)/Ui∗ . Then G is a pseudofinite group definable in M∗, and thus is fsg(see Example 6.2). Since G is a definable quotient of K(M∗), and K(M∗)/K(M∗)00

is profinite by Fact 7.2, it follows (using Fact 2.3(a, b)) that G/G00 is also profinite.By Theorem 6.6, there are m,n, r ∈ N, with n ≥ 1, m < ǫn, and r ≤ n, and aformula φ(x; y) (with no parameters) such that:

(∗) For some b ∈ (M∗)y , H := φ(M∗; b) is a normal subgroup of G = K(M∗)/Ui∗

of index n, and there is Z ⊆ G, which is a union of m cosets of H such that(A(M∗)/Ui∗)\Z is a union of r cosets of H .

In particular, (∗) is a property of i∗ that can be expressed using a formula ζ(t)over Qp. So there is n′ ≥ n such that Qan

p |= ζ(in′), which is a contradiction. �

Remark 7.6. Suppose K is a compact p-adic analytic group such that the family ofopen normal subgroups is (eventually) linearly ordered by inclusion (e.g., K = Zp).In this case, Theorem 7.5 can be deduced easily from Theorem 7.3. Indeed, givena definable set A ⊆ G and some ǫ > 0, let H ≤ K and Z ⊆ K be as in Theorem7.3. Then all but finitely many open normal subgroups of K are contained in Hand, given such a subgroup Ui ≤ H , if Hi = H/Ui and Zi = Z/Ui, then Hi andZi satisfy conditions (i) and (ii) in Theorem 7.5. Choosing n sufficiently large, wecan then let Hi be trivial for any Ui not contained in H . It is not clear whethersuch an argument can be given for arbitrary K (or if there is a proof of Theorem7.5 not relying on Fact 7.1).

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DPMMS, University of Cambridge, Cambridge CB3 0WB, UK

E-mail address: [email protected]

Department of Mathematics, University of Notre Dame, Notre Dame IN 46656, USA


Department of Mathematics, University of Chicago, Chicago IL 60637, USA




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