Pseudofinite groups and tame arithmetic regularity Gabriel Conant - - PowerPoint PPT Presentation

Pseudofinite groups and tame arithmetic regularity

Gabriel Conant

Notre Dame

23 July 2018 Logic Colloquium University of Udine

Regularity

Szemer´ edi’s Regularity Lemma (1976): Any sufficiently large finite graph can be nontrivially partitioned into a small number of pieces so that most pairs of pieces have regular edge distribution.

Regularity

Szemer´ edi’s Regularity Lemma (1976): Any sufficiently large finite graph can be nontrivially partitioned into a small number of pieces so that most pairs of pieces have regular edge distribution. Green (2005): For any A ⊆ (Z/pZ)n, there is a subgroup H of small index such that A is uniformly distributed inside almost all cosets of H.

Regularity

Szemer´ edi’s Regularity Lemma (1976): Any sufficiently large finite graph can be nontrivially partitioned into a small number of pieces so that most pairs of pieces have regular edge distribution. Green (2005): For any A ⊆ (Z/pZ)n, there is a subgroup H of small index such that A is uniformly distributed inside almost all cosets of H. This is a special case of Green’s “arithmetic regularity lemma for finite abelian groups”.

Regularity

Szemer´ edi’s Regularity Lemma (1976): Any sufficiently large finite graph can be nontrivially partitioned into a small number of pieces so that most pairs of pieces have regular edge distribution. Green (2005): For any A ⊆ (Z/pZ)n, there is a subgroup H of small index such that A is uniformly distributed inside almost all cosets of H. This is a special case of Green’s “arithmetic regularity lemma for finite abelian groups”. ← − highly structured highly random − →

Tame arithmetic regularity

• Terry, Wolf (5 Oct 2017) stable sets in (Z/pZ)n (quantitative)

Tame arithmetic regularity

• Terry, Wolf (5 Oct 2017) stable sets in (Z/pZ)n (quantitative)
• C., Pillay, Terry (17 Oct 2017) stable sets in finite groups (qualitative)

Tame arithmetic regularity

• Terry, Wolf (5 Oct 2017) stable sets in (Z/pZ)n (quantitative)
• C., Pillay, Terry (17 Oct 2017) stable sets in finite groups (qualitative)
• Alon, Fox, Zhao (15 Jan 2018) VC-sets in finite abelian groups of

bounded exponent (quantitative)

Tame arithmetic regularity

• Terry, Wolf (5 Oct 2017) stable sets in (Z/pZ)n (quantitative)
• C., Pillay, Terry (17 Oct 2017) stable sets in finite groups (qualitative)
• Alon, Fox, Zhao (15 Jan 2018) VC-sets in finite abelian groups of

bounded exponent (quantitative)

• Sisask (8 Feb 2018) VC-sets in finite abelian groups (quantitative)

Tame arithmetic regularity

• Terry, Wolf (5 Oct 2017) stable sets in (Z/pZ)n (quantitative)
• C., Pillay, Terry (17 Oct 2017) stable sets in finite groups (qualitative)
• Alon, Fox, Zhao (15 Jan 2018) VC-sets in finite abelian groups of

bounded exponent (quantitative)

• Sisask (8 Feb 2018) VC-sets in finite abelian groups (quantitative)
• C., Pillay, Terry (12 Feb 2018) VC-sets in finite groups, and in finite

groups of bounded exponent (qualitative)

Tame arithmetic regularity

• Terry, Wolf (5 Oct 2017) stable sets in (Z/pZ)n (quantitative)
• C., Pillay, Terry (17 Oct 2017) stable sets in finite groups (qualitative)
• Alon, Fox, Zhao (15 Jan 2018) VC-sets in finite abelian groups of

bounded exponent (quantitative)

• Sisask (8 Feb 2018) VC-sets in finite abelian groups (quantitative)
• C., Pillay, Terry (12 Feb 2018) VC-sets in finite groups, and in finite

groups of bounded exponent (qualitative)

• Terry, Wolf (17 May 2018) stable sets in finite abelian groups

(quantitative)

Tame arithmetic regularity

• Terry, Wolf (5 Oct 2017) stable sets in (Z/pZ)n (quantitative)
• C., Pillay, Terry (17 Oct 2017) stable sets in finite groups (qualitative)
• Alon, Fox, Zhao (15 Jan 2018) VC-sets in finite abelian groups of

bounded exponent (quantitative)

• Sisask (8 Feb 2018) VC-sets in finite abelian groups (quantitative)
• C., Pillay, Terry (12 Feb 2018) VC-sets in finite groups, and in finite

groups of bounded exponent (qualitative)

• Terry, Wolf (17 May 2018) stable sets in finite abelian groups

(quantitative)

• C. (15 June 2018) VC-sets in finite groups of bounded exponent

(99% quantitative), plus some further results

VC-sets in groups

Given a group G and a subset A ⊆ G, let VC(A) denote the VC-dimension of {gA : g ∈ G}. In other words, VC(A) ≥ d if and only if there is X ⊆ G such that |X| = d and {X ∩ gA : g ∈ G} = P(X).

VC-sets in groups

Given a group G and a subset A ⊆ G, let VC(A) denote the VC-dimension of {gA : g ∈ G}. In other words, VC(A) ≥ d if and only if there is X ⊆ G such that |X| = d and {X ∩ gA : g ∈ G} = P(X). Remark: VC(A) is finite if and only if the formula xy ∈ A is NIP in the structure (G, ·, A).

VC-sets in finite abelian groups of bounded exponent

Theorem (Alon, Fox, Zhao 2018)

Suppose G is a finite abelian group of exponent at most r and A ⊆ G is such that VC(A) ≤ d.

VC-sets in finite abelian groups of bounded exponent

Theorem (Alon, Fox, Zhao 2018)

Suppose G is a finite abelian group of exponent at most r and A ⊆ G is such that VC(A) ≤ d. Then, for any ǫ > 0, there are: ∗ a subgroup H ≤ G, of index Or,d((1/ǫ)d+1), and ∗ a set D ⊆ G, which is a union of cosets of H, such that |A △D| ≤ ǫ|G|.

VC-sets in finite abelian groups of bounded exponent

Theorem (Alon, Fox, Zhao 2018)

Suppose G is a finite abelian group of exponent at most r and A ⊆ G is such that VC(A) ≤ d. Then, for any ǫ > 0, there are: ∗ a subgroup H ≤ G, of index Or,d((1/ǫ)d+1), and ∗ a set D ⊆ G, which is a union of cosets of H, such that |A △D| ≤ ǫ|G|. Given a finite group G, a subset A ⊆ G, and ǫ > 0, define Stabǫ(A) = {x ∈ G : |xA △A| ≤ ǫ|G|}.

VC-sets in finite abelian groups of bounded exponent

Theorem (Alon, Fox, Zhao 2018)

Suppose G is a finite abelian group of exponent at most r and A ⊆ G is such that VC(A) ≤ d. Then, for any ǫ > 0, there are: ∗ a subgroup H ≤ G, of index Or,d((1/ǫ)d+1), and ∗ a set D ⊆ G, which is a union of cosets of H, such that |A △D| ≤ ǫ|G|. Given a finite group G, a subset A ⊆ G, and ǫ > 0, define Stabǫ(A) = {x ∈ G : |xA △A| ≤ ǫ|G|}. Idea: In abelian groups of bounded exponent, stabilizers of VC-sets contain large subgroups.

Ingredients of the proof

Let G be a finite group.

Ingredients of the proof

Let G be a finite group.

Corollary of Haussler’s Packing Lemma

If A ⊆ G and VC(A) ≤ d, then | Stabǫ(A)| ≥ (ǫ/30)d|G|.

Ingredients of the proof

Let G be a finite group.

Corollary of Haussler’s Packing Lemma

If A ⊆ G and VC(A) ≤ d, then | Stabǫ(A)| ≥ (ǫ/30)d|G|.

Bogolyubov-Ruzsa Lemma (bounded exponent case)

Assume G is abelian of exponent r. Fix a nonempty set S ⊆ G, with |S + S| ≤ k|S|. Then 2S − 2S contains a subgroup H of size Ωr,k(|S|).

Ingredients of the proof

Let G be a finite group.

Corollary of Haussler’s Packing Lemma

If A ⊆ G and VC(A) ≤ d, then | Stabǫ(A)| ≥ (ǫ/30)d|G|.

Bogolyubov-Ruzsa Lemma (bounded exponent case)

Assume G is abelian of exponent r. Fix a nonempty set S ⊆ G, with |S + S| ≤ k|S|. Then 2S − 2S contains a subgroup H of size Ωr,k(|S|). Note: If S ⊆ G and |S| ≥ α|G|, then |S + S| ≤ |G| ≤ α-1|S|.

Ingredients of the proof

Let G be a finite group.

Corollary of Haussler’s Packing Lemma

If A ⊆ G and VC(A) ≤ d, then | Stabǫ(A)| ≥ (ǫ/30)d|G|.

Bogolyubov-Ruzsa Lemma (bounded exponent case)

Assume G is abelian of exponent r. Fix a nonempty set S ⊆ G, with |S + S| ≤ k|S|. Then 2S − 2S contains a subgroup H of size Ωr,k(|S|). Note: If S ⊆ G and |S| ≥ α|G|, then |S + S| ≤ |G| ≤ α-1|S|.

Lemma (Alon, Fox, Zhao)

Fix A ⊆ G and suppose H ≤ G is contained in Stabǫ(A). Then there is D ⊆ G, which is a union of right cosets of H, such that |A △D| ≤ ǫ|G|.

Approximate subgroups

Fix a group G and a nonempty finite subset S ⊆ G. Definition: S is a k-approximate subgroup if 1 ∈ S, S = S-1, and S2 is covered by k left translates of S.

Approximate subgroups

Fix a group G and a nonempty finite subset S ⊆ G. Definition: S is a k-approximate subgroup if 1 ∈ S, S = S-1, and S2 is covered by k left translates of S.

Theorem (Breuillard, Green, Tao; Hrushovski)

Suppose G has exponent r and S is a k-approximate subgroup. Then S4 contains a subgroup H of size Ωr,k(|S|).

Approximate subgroups

Fix a group G and a nonempty finite subset S ⊆ G. Definition: S is a k-approximate subgroup if 1 ∈ S, S = S-1, and S2 is covered by k left translates of S.

Theorem (Breuillard, Green, Tao; Hrushovski)

Suppose G has exponent r and S is a k-approximate subgroup. Then S4 contains a subgroup H of size Ωr,k(|S|).

Theorem (Tao)

If |S3| ≤ k|S| then (S ∪ S-1)2 is a O(kO(1))-approximate subgroup.

Corollary (weak Bogolyubov-Ruzsa)

Suppose G has exponent r, S = S-1, and |S3| ≤ k|S|. Then S8 contains a subgroup of size Ωr,k(|S|).

VC-sets in abelian groups of bounded exponent

Theorem (C. 2018)

Suppose G is a finite group of exponent at most r and A ⊆ G is such that VC(A) ≤ d. Then, for any ǫ > 0, there are: ∗ a subgroup H ≤ G, of index Or,d((1/ǫ)d+1), and ∗ a set D ⊆ G, which is a union of right cosets of H, such that |A △D| ≤ ǫ|G|.

VC-sets in abelian groups of bounded exponent

Theorem (C. 2018)

Suppose G is a finite group of exponent at most r and A ⊆ G is such that VC(A) ≤ d. Then, for any ǫ > 0, there are: ∗ a subgroup H ≤ G, of index Or,d((1/ǫ)d+1), and ∗ a set D ⊆ G, which is a union of right cosets of H, such that |A △D| ≤ ǫ|G|. Recall: If H ≤ G and K =

g∈G gHg-1 then [G : K] ≤ [G : H]!.

VC-sets in abelian groups of bounded exponent

Theorem (C. 2018)

Suppose G is a finite group of exponent at most r and A ⊆ G is such that VC(A) ≤ d. Then, for any ǫ > 0, there are: ∗ a normal subgroup H ≤ G, of index 2Or,d((1/ǫ)d+1), and ∗ a set D ⊆ G, which is a union of cosets of H, such that |A △D| ≤ ǫ|G|.

VC-sets in abelian groups of bounded exponent

Theorem (C. 2018)

Suppose G is a finite group of exponent at most r and A ⊆ G is such that VC(A) ≤ d. Then, for any ǫ > 0, there are: ∗ a normal subgroup H ≤ G, of index 2Or,d((1/ǫ)d+1), and ∗ a set D ⊆ G, which is a union of cosets of H, such that |A △D| ≤ ǫ|G|. Moreover, there is a set Z ⊆ G, with |Z| ≤ ǫ1/2|G|, such that for any g ∈ Z, either |gH ∩ A| ≤ ǫ1/4|H| or |gH ∩ A| ≥ (1 − ǫ1/4)|H|.

VC-sets in abelian groups of bounded exponent

Theorem (C. 2018)

Suppose G is a finite group of exponent at most r and A ⊆ G is such that VC(A) ≤ d. Then, for any ǫ > 0, there are: ∗ a normal subgroup H ≤ G, of index 2Or,d((1/ǫ)d+1), and ∗ a set D ⊆ G, which is a union of cosets of H, such that |A △D| ≤ ǫ|G|. Moreover, there is a set Z ⊆ G, with |Z| ≤ ǫ1/2|G|, such that for any g ∈ Z, either |gH ∩ A| ≤ ǫ1/4|H| or |gH ∩ A| ≥ (1 − ǫ1/4)|H|. This yields a regular partition for the bipartite graph on (G, G) induced by xy ∈ A, in which the pieces are the cosets of H and the regular pairs have density within ǫ of 0 or 1.

Removing the bound on the exponent

AFZ: If G = (Z/pZ, +) and A = {0, 1, . . . , p−1

2 } then VC(A) ≤ 3.

Removing the bound on the exponent

AFZ: If G = (Z/pZ, +) and A = {0, 1, . . . , p−1

2 } then VC(A) ≤ 3.

But we cannot have |A △D| < 1

2|G|, where D is a union of cosets of a

subgroup of Z/pZ whose index is independent of p.

Bohr sets

Definition

Given a group H, a homomorphism τ : H → Tn, and some δ > 0, set Bn

δ,τ := {x ∈ H : d(τ(x), 0) < δ},

where d denotes the usual metric on Tn, and 0 is the identity in Tn.

Bohr sets

Definition

Given a group H, a homomorphism τ : H → Tn, and some δ > 0, set Bn

δ,τ := {x ∈ H : d(τ(x), 0) < δ},

where d denotes the usual metric on Tn, and 0 is the identity in Tn. B ⊆ H is a (δ, n)-Bohr set in H if B = Bn

τ,δ for some τ.

Bohr sets

Definition

Given a group H, a homomorphism τ : H → Tn, and some δ > 0, set Bn

δ,τ := {x ∈ H : d(τ(x), 0) < δ},

where d denotes the usual metric on Tn, and 0 is the identity in Tn. B ⊆ H is a (δ, n)-Bohr set in H if B = Bn

τ,δ for some τ.

Remarks:

• Bn

δ,τ is closed under inverses and contains the identity.

Bohr sets

Definition

Given a group H, a homomorphism τ : H → Tn, and some δ > 0, set Bn

δ,τ := {x ∈ H : d(τ(x), 0) < δ},

where d denotes the usual metric on Tn, and 0 is the identity in Tn. B ⊆ H is a (δ, n)-Bohr set in H if B = Bn

τ,δ for some τ.

Remarks:

• Bn

δ,τ is closed under inverses and contains the identity.

• Bn

δ,τ · Bn δ,τ ⊆ Bn 2δ,τ.

Bohr sets

Definition

Given a group H, a homomorphism τ : H → Tn, and some δ > 0, set Bn

δ,τ := {x ∈ H : d(τ(x), 0) < δ},

where d denotes the usual metric on Tn, and 0 is the identity in Tn. B ⊆ H is a (δ, n)-Bohr set in H if B = Bn

τ,δ for some τ.

Remarks:

• Bn

δ,τ is closed under inverses and contains the identity.

• Bn

δ,τ · Bn δ,τ ⊆ Bn 2δ,τ.

• Bohr sets in abelian groups contain large “coset progressions”.

Dense subsets of finite groups

Bogolyubov’s Lemma (1939; see Ruzsa 1994)

Suppose G is a finite abelian group and S ⊆ G is such that |S| ≥ α|G|. Then 2S − 2S contains a (1/4, n)-Bohr set B ⊆ G, with n < (1/α)2.

Dense subsets of finite groups

Bogolyubov’s Lemma (1939; see Ruzsa 1994)

Suppose G is a finite abelian group and S ⊆ G is such that |S| ≥ α|G|. Then 2S − 2S contains a (1/4, n)-Bohr set B ⊆ G, with n < (1/α)2.

Theorem (C. 2018)

Suppose G is a finite group and S ⊆ G is such that |S| ≥ α|G|. Then there are: ∗ a normal subgroup H ≤ G of index Oα(1), and ∗ a (δ, n)-Bohr set B ⊆ H, with δ-1, n ≤ Oα(1), such that B ⊆ S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2.

Dense subsets of finite groups

Bogolyubov’s Lemma (1939; see Ruzsa 1994)

Suppose G is a finite abelian group and S ⊆ G is such that |S| ≥ α|G|. Then 2S − 2S contains a (1/4, n)-Bohr set B ⊆ G, with n < (1/α)2.

Theorem (C. 2018)

Suppose G is a finite group and S ⊆ G is such that |S| ≥ α|G|. Then there are: ∗ a normal subgroup H ≤ G of index Oα(1), and ∗ a (δ, n)-Bohr set B ⊆ H, with δ-1, n ≤ Oα(1), such that B ⊆ S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2. The proof uses pseudofinite groups, specifically the “ultraproduct of counterexamples” method.

Pseudofinite ingredients

Let G be a saturated expansion of a group, and assume Th(G) is

• pseudofinite. Suppose S ⊆ G is definable and µ(S) > 0, where µ is

the normalized pseudofinite counting measure.

Pseudofinite ingredients

Let G be a saturated expansion of a group, and assume Th(G) is

• pseudofinite. Suppose S ⊆ G is definable and µ(S) > 0, where µ is

the normalized pseudofinite counting measure.

• “Pseudofinite Sanders-Croot-Sisask analysis”: There is a countably

type-definable, bounded-index, normal subgroup K ≤ G such that K ⊆ S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2. (Similar to Massicot-Wagner and Krupi´ nski-Pillay.)

Pseudofinite ingredients

Let G be a saturated expansion of a group, and assume Th(G) is

• pseudofinite. Suppose S ⊆ G is definable and µ(S) > 0, where µ is

the normalized pseudofinite counting measure.

• “Pseudofinite Sanders-Croot-Sisask analysis”: There is a countably

type-definable, bounded-index, normal subgroup K ≤ G such that K ⊆ S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2. (Similar to Massicot-Wagner and Krupi´ nski-Pillay.)

• Pillay: The connected component of the identity in G/K is abelian.

(Nikolov, Schneider, Thom: True for any compactification of G.)

Pseudofinite ingredients

Let G be a saturated expansion of a group, and assume Th(G) is

• pseudofinite. Suppose S ⊆ G is definable and µ(S) > 0, where µ is

the normalized pseudofinite counting measure.

• “Pseudofinite Sanders-Croot-Sisask analysis”: There is a countably

type-definable, bounded-index, normal subgroup K ≤ G such that K ⊆ S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2. (Similar to Massicot-Wagner and Krupi´ nski-Pillay.)

• Pillay: The connected component of the identity in G/K is abelian.

(Nikolov, Schneider, Thom: True for any compactification of G.)

• Structure of compact groups: K = ∞

i=0 Bi, where Bi is a

(δi, ni)-Bohr set in a definable finite-index normal Hi ≤ G.

Pseudofinite ingredients

Let G be a saturated expansion of a group, and assume Th(G) is

• pseudofinite. Suppose S ⊆ G is definable and µ(S) > 0, where µ is

the normalized pseudofinite counting measure.

• “Pseudofinite Sanders-Croot-Sisask analysis”: There is a countably

type-definable, bounded-index, normal subgroup K ≤ G such that K ⊆ S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2. (Similar to Massicot-Wagner and Krupi´ nski-Pillay.)

• Pillay: The connected component of the identity in G/K is abelian.

(Nikolov, Schneider, Thom: True for any compactification of G.)

• Structure of compact groups: K = ∞

i=0 Bi, where Bi is a

(δi, ni)-Bohr set in a definable finite-index normal Hi ≤ G.

• C.-Pillay-Terry: Replace Bi by a definable “approximate Bohr set”.

Pseudofinite ingredients

Let G be a saturated expansion of a group, and assume Th(G) is

• pseudofinite. Suppose S ⊆ G is definable and µ(S) > 0, where µ is

the normalized pseudofinite counting measure.

• “Pseudofinite Sanders-Croot-Sisask analysis”: There is a countably

type-definable, bounded-index, normal subgroup K ≤ G such that K ⊆ S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2. (Similar to Massicot-Wagner and Krupi´ nski-Pillay.)

• Pillay: The connected component of the identity in G/K is abelian.

(Nikolov, Schneider, Thom: True for any compactification of G.)

• Structure of compact groups: K = ∞

i=0 Bi, where Bi is a

(δi, ni)-Bohr set in a definable finite-index normal Hi ≤ G.

• C.-Pillay-Terry: Replace Bi by a definable “approximate Bohr set”.
• Alekseev, Glebskiˇ

ı, Gordon: Approximate Bohr sets in finite groups contain large Bohr sets.

The abelian exponent of a group

The abelian exponent of a group G is the smallest r ≥ 1 (if it exists) such that H/[H, H] has exponent at most r for any normal H ≤ G.

The abelian exponent of a group

The abelian exponent of a group G is the smallest r ≥ 1 (if it exists) such that H/[H, H] has exponent at most r for any normal H ≤ G. Facts:

• The abelian exponent of G is at most the exponent of G.

The abelian exponent of a group

The abelian exponent of a group G is the smallest r ≥ 1 (if it exists) such that H/[H, H] has exponent at most r for any normal H ≤ G. Facts:

• The abelian exponent of G is at most the exponent of G.
• If G is nonabelian and simple then it has abelian exponent 1.

The abelian exponent of a group

The abelian exponent of a group G is the smallest r ≥ 1 (if it exists) such that H/[H, H] has exponent at most r for any normal H ≤ G. Facts:

• The abelian exponent of G is at most the exponent of G.
• If G is nonabelian and simple then it has abelian exponent 1.
• If G has abelian exponent r and H has abelian exponent 1 then

G × H has abelian exponent r.

Bogolyubov’s Lemma for bounded abelian exponent

Corollary

Suppose G is a finite group of abelian exponent r, and S ⊆ G is such that |S| ≥ α|G|. Then S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2 contains a normal subgroup of G of index Or,α(1).

Bogolyubov’s Lemma for bounded abelian exponent

Corollary

Suppose G is a finite group of abelian exponent r, and S ⊆ G is such that |S| ≥ α|G|. Then S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2 contains a normal subgroup of G of index Or,α(1). Proof: Let S∗ = S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2.

Bogolyubov’s Lemma for bounded abelian exponent

Corollary

Suppose G is a finite group of abelian exponent r, and S ⊆ G is such that |S| ≥ α|G|. Then S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2 contains a normal subgroup of G of index Or,α(1). Proof: Let S∗ = S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2.

• We have B ⊆ S∗, where B is a (δ, n)-Bohr set in a normal subgroup

H ≤ G, with δ-1, n, [G : H] ≤ Oα(1).

Bogolyubov’s Lemma for bounded abelian exponent

Corollary

Suppose G is a finite group of abelian exponent r, and S ⊆ G is such that |S| ≥ α|G|. Then S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2 contains a normal subgroup of G of index Or,α(1). Proof: Let S∗ = S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2.

• We have B ⊆ S∗, where B is a (δ, n)-Bohr set in a normal subgroup

H ≤ G, with δ-1, n, [G : H] ≤ Oα(1).

• B contains the kernel K of some homomorphism τ : H → Tn.

Bogolyubov’s Lemma for bounded abelian exponent

Corollary

Suppose G is a finite group of abelian exponent r, and S ⊆ G is such that |S| ≥ α|G|. Then S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2 contains a normal subgroup of G of index Or,α(1). Proof: Let S∗ = S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2.

• We have B ⊆ S∗, where B is a (δ, n)-Bohr set in a normal subgroup

H ≤ G, with δ-1, n, [G : H] ≤ Oα(1).

• B contains the kernel K of some homomorphism τ : H → Tn.
• H/K ≤ Tn is a finite abelian group of exponent at most r, and is

generated by at most n elements. So |H/K| ≤ r n ≤ Or,α(1).

Bogolyubov’s Lemma for bounded abelian exponent

Corollary

Suppose G is a finite group of abelian exponent r, and S ⊆ G is such that |S| ≥ α|G|. Then S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2 contains a normal subgroup of G of index Or,α(1). Proof: Let S∗ = S2S-2 ∩ (SS-1)2 ∩ S-2S2 ∩ (S-1S)2.

• We have B ⊆ S∗, where B is a (δ, n)-Bohr set in a normal subgroup

H ≤ G, with δ-1, n, [G : H] ≤ Oα(1).

• B contains the kernel K of some homomorphism τ : H → Tn.
• H/K ≤ Tn is a finite abelian group of exponent at most r, and is

generated by at most n elements. So |H/K| ≤ r n ≤ Or,α(1).

• Now

g∈G gKg-1 is the desired normal subgroup of G.

VC-sets and abelian exponent

Theorem (C. 2018)

Suppose G is a finite group of abelian exponent at most r and A ⊆ G is such that VC(A) ≤ d. Then, for any ǫ > 0, there are: ∗ a normal subgroup H ≤ G, of index Or,d,ǫ(1), ∗ a set D ⊆ G, which is a union of cosets of H, and ∗ a set Z ⊆ G, with |Z| ≤ ǫ1/2|G|, such that (i) |A △D| ≤ ǫ|G|, and (ii) for any g ∈ Z, either |gH ∩ A| ≤ ǫ1/4|G| or |gH\A| ≤ ǫ1/4|G|.

VC sets in nonabelian finite simple groups

Corollary

For any d ≥ 1 and ǫ > 0, there is n = n(d, ǫ) such that if G is a nonabelian finite simple group of size at least n, and A ⊆ G is such that VC(A) ≤ d, then |A| < ǫ|G| or |A| > (1 − ǫ)|G|.

VC sets in nonabelian finite simple groups

Corollary

For any d ≥ 1 and ǫ > 0, there is n = n(d, ǫ) such that if G is a nonabelian finite simple group of size at least n, and A ⊆ G is such that VC(A) ≤ d, then |A| < ǫ|G| or |A| > (1 − ǫ)|G|. Using work of Gowers on “quasirandom” groups, one can show n(d, ǫ) = 2O((90/ǫ)6d).

thank you