theorem GrowthInGroups.Lecture2.lemma_4_2 {G : Type u_1} [DecidableEq G] [Group G] (U V W : Finset G) : U.card * (V⁻¹ * W).card ≤ (U * V).card * (U * W).card

theorem GrowthInGroups.Lecture2.lemma_4_3_2 {G : Type u_1} [DecidableEq G] [Group G] {A : Finset G} {K : ℝ} (hA : ↑(A ^ 2).card ≤ K * ↑A.card) : ↑(A⁻¹ * A).card ≤ K ^ 2 * ↑A.card

theorem GrowthInGroups.Lecture2.lemma_4_3_1 {G : Type u_1} [DecidableEq G] [Group G] {A : Finset G} {K : ℝ} (hA : ↑(A ^ 2).card ≤ K * ↑A.card) : ↑(A * A⁻¹).card ≤ K ^ 2 * ↑A.card

theorem GrowthInGroups.Lecture2.lemma_4_4_1 {G : Type u_1} [DecidableEq G] [Group G] {A : Finset G} {K : ℝ} {m : ℕ} (hm : 3 ≤ m) (hA : ↑(A ^ 3).card ≤ K * ↑A.card) (ε : Fin m → ℤ) (hε : ∀ (i : Fin m), |ε i| = 1) : ↑(List.map (fun (i : Fin m) => A ^ ε i) (List.finRange m)).prod.card ≤ K ^ (3 * (m - 2)) * ↑A.card

theorem GrowthInGroups.Lecture2.lemma_4_4_2 {G : Type u_1} [DecidableEq G] [Group G] {A : Finset G} {K : ℝ} {m : ℕ} (hm : 3 ≤ m) (hA : ↑(A ^ 3).card ≤ K * ↑A.card) (hAsymm : A⁻¹ = A) : ↑(A ^ m).card ≤ K ^ (m - 2) * ↑A.card

def GrowthInGroups.Lecture2.def_4_5 {G : Type u_1} [Group G] (S : Set G) (K : ℝ) : Prop

Equations

• GrowthInGroups.Lecture2.def_4_5 S K = IsApproximateSubgroup K S

theorem GrowthInGroups.Lecture2.remark_4_6_1 (k : ℕ) : IsApproximateAddSubgroup 2 (Set.Icc (-↑k) ↑k)

theorem GrowthInGroups.Lecture2.remark_4_6_2 {ι : Type u_2} [Fintype ι] (k : ι → ℕ) : IsApproximateAddSubgroup (2 ^ Fintype.card ι) (Set.univ.pi fun (i : ι) => Set.Icc (-↑(k i)) ↑(k i))

theorem GrowthInGroups.Lecture2.remark_4_6_3 : IsApproximateAddSubgroup 2 (Set.Icc (-1) 1)

theorem GrowthInGroups.Lecture2.lemma_4_7 {G : Type u_1} [DecidableEq G] [Group G] {K : ℝ} {A : Finset G} (hA₁ : 1 ∈ A) (hsymm : A⁻¹ = A) (hA : ↑(A ^ 3).card ≤ K * ↑A.card) : IsApproximateSubgroup (K ^ 3) (↑A ^ 2)

theorem GrowthInGroups.Lecture2.lemma_4_8 {G : Type u_1} [DecidableEq G] [Group G] {K : ℝ} {A B : Finset G} (hB : B.Nonempty) (hK : ↑(A * B).card ≤ K * ↑B.card) : ∃ F ⊆ A, ↑F.card ≤ K ∧ A ⊆ F * B / B