| import Mathlib |
| import OAI.Probability.CriticalPercolation.Growth.CosetBallIndex |
| import OAI.Probability.CriticalPercolation.Growth.CosetSparseGrowth |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical Pointwise |
|
|
| namespace CriticalPercolation.RelativeGrowth |
| variable {A : Type*} [Group A] |
|
|
| def Bounded (S T : Set A) : Prop := ∃ n : ℕ, T⊆S^n |
| lemma bounded_singleton (S : Set A) (h1 : (1:A)∈S) (hi : S⁻¹=S) |
| (hg : Subgroup.closure S=⊤) (a : A) : Bounded S {a} := by |
| have ha : a∈Subgroup.closure S := hg.symm ▸ Subgroup.mem_top a |
| suffices ∃ n, a∈S^n by simpa [Bounded] using this |
| induction ha using Subgroup.closure_induction with |
| | mem a ha => exact ⟨1,by simpa using ha⟩ |
| | one => exact ⟨1,by simpa using h1⟩ |
| | mul a b _ _ ha hb => |
| obtain ⟨n,hn⟩ := ha; obtain ⟨m,hm⟩ := hb |
| exact ⟨n+m,by rw [pow_add]; exact Set.mul_mem_mul hn hm⟩ |
| | inv a _ ha => |
| obtain ⟨n,hn⟩ := ha |
| refine ⟨n,?_⟩ |
| rw [←hi,inv_pow] |
| exact Set.inv_mem_inv.mpr hn |
| lemma bounded_finite (S : Set A) (h1 : (1:A)∈S) (hi : S⁻¹=S) |
| (hg : Subgroup.closure S=⊤) {F : Set A} (hf : F.Finite) : Bounded S F := by |
| have hh : ∀ f∈F, ∃ n, f∈S^n := by |
| intro f hf |
| obtain ⟨n,hn⟩ := bounded_singleton S h1 hi hg f |
| exact ⟨n,hn (Set.mem_singleton _)⟩ |
| choose n hn using hh |
| let L := hf.toFinset.sup (fun a => if ha : a∈F then n a ha else 0) |
| refine ⟨L,?_⟩ |
| intro a ha |
| apply Set.pow_subset_pow_right h1 (show n a ha≤L from ?_) (hn a ha) |
| have hh := Finset.le_sup (f:=fun a => if ha : a∈F then n a ha else 0) (hf.mem_toFinset.mpr ha) |
| simpa only [dite_eq_left ha] using hh |
| lemma Bounded.mono {S T U : Set A} (h : Bounded S T) (hUT : U⊆T) : Bounded S U := |
| ⟨h.choose,hUT.trans h.choose_spec⟩ |
| lemma Bounded.mul {S T U : Set A} (hT : Bounded S T) (hU : Bounded S U) : Bounded S (T*U) := by |
| obtain ⟨n,hn⟩ := hT; obtain ⟨m,hm⟩ := hU |
| exact ⟨n+m,by rw [pow_add]; exact Set.mul_subset_mul hn hm⟩ |
| lemma Bounded.inv {S T : Set A} (hi : S⁻¹=S) (hT : Bounded S T) : Bounded S T⁻¹ := by |
| obtain ⟨n,hn⟩ := hT |
| refine ⟨n,?_⟩ |
| rw [←hi,inv_pow] |
| exact Set.inv_subset_inv.mpr hn |
| lemma Bounded.union {S T U : Set A} (h1 : (1:A)∈S) |
| (hT : Bounded S T) (hU : Bounded S U) : Bounded S (T∪U) := by |
| obtain ⟨n,hn⟩ := hT; obtain ⟨m,hm⟩ := hU |
| exact ⟨max n m,Set.union_subset (hn.trans (Set.pow_subset_pow_right h1 (le_max_left _ _))) |
| (hm.trans (Set.pow_subset_pow_right h1 (le_max_right _ _)))⟩ |
|
|
| lemma finite_coset_image_iff_cover (K : Subgroup A) (T : Set A) : |
| ((QuotientGroup.mk : A → A ⧸ K) '' T).Finite ↔ |
| ∃ F : Set A, F.Finite ∧ T⊆F*(K:Set A) := by |
| constructor |
| · intro hf |
| let Q := (QuotientGroup.mk : A → A ⧸ K) '' T |
| have hh : ∀ q∈Q, ∃ a∈T, (QuotientGroup.mk a : A ⧸ K)=q := fun q hq => hq |
| choose a ha he using hh |
| let F := Set.range (fun q : Q => a q.val q.property) |
| let : Finite Q := hf.to_subtype |
| refine ⟨F,Set.finite_range _,?_⟩ |
| intro t ht |
| let q : Q := ⟨QuotientGroup.mk t,⟨t,ht,rfl⟩⟩ |
| refine ⟨a q.val q.property,⟨q,rfl⟩,(a q.val q.property)⁻¹*t,?_,by simp⟩ |
| exact QuotientGroup.eq.mp (he q.val q.property) |
| · rintro ⟨F,hF,hcov⟩ |
| apply (hF.image (QuotientGroup.mk : A → A ⧸ K)).subset |
| rintro _ ⟨t,ht,rfl⟩ |
| obtain ⟨f,hf,k,hk,rfl⟩ := hcov ht |
| exact ⟨f,hf,QuotientGroup.eq.mpr (by simpa using hk)⟩ |
|
|
| lemma bounded_of_finite_coset_image (S : Set A) (K : Subgroup A) |
| (h1 : (1:A)∈S) (hi : S⁻¹=S) (hg : Subgroup.closure S=⊤) (hK : Bounded S (K:Set A)) |
| (T : Set A) (hf : ((QuotientGroup.mk : A → A ⧸ K) '' T).Finite) : Bounded S T := by |
| obtain ⟨F,hF,hcov⟩ := (finite_coset_image_iff_cover K T).mp hf |
| exact ((bounded_finite S h1 hi hg hF).mul hK).mono hcov |
| lemma finite_coset_image_of_bounded (K : Subgroup A) (S T : Set A) |
| (hf : ∀ n, (ball K S n).Finite) (hT : Bounded S T) : |
| ((QuotientGroup.mk : A → A ⧸ K) '' T).Finite := by |
| obtain ⟨n,hn⟩ := hT |
| exact (hf n).subset (Set.image_mono hn) |
|
|
| lemma HasDegree.bounded_generators {K : Subgroup A} {S T : Set A} {d : ℕ} |
| (h1 : (1:A)∈S) (hg : HasDegree K S d) (hT : Bounded S T) : HasDegree K T d := by |
| obtain ⟨L,hL⟩ := hT |
| have hin (n : ℕ) : T^n⊆S^((L+1)*n) := by |
| rw [pow_mul] |
| exact Set.pow_subset_pow_left (hL.trans (Set.pow_subset_pow_right h1 (Nat.le_succ L))) |
| refine ⟨fun n => (hg.1 _).subset (Set.image_mono (hin n)),?_⟩ |
| obtain ⟨C,hC⟩ := hg.2 |
| refine ⟨C*(2*(L+1))^d,?_⟩ |
| intro R |
| obtain ⟨r,hr,hbound⟩ := hC ((L+1)*max R 1) |
| let n := r/(L+1) |
| have hn : max R 1≤n := (Nat.le_div_iff_mul_le (by omega : 0<L+1)).mpr (by simpa [Nat.mul_comm] using hr) |
| have hnr : (L+1)*n≤r := by simpa only [n,Nat.mul_comm] using Nat.div_mul_le_self r (L+1) |
| have hrn : r≤(2*(L+1))*n := by |
| have hlt := Nat.lt_mul_div_succ r (by omega : 0<L+1) |
| change r<(L+1)*(n+1) at hlt |
| have hh : 1≤n := (le_max_right R 1).trans hn |
| nlinarith |
| refine ⟨n,(le_max_left R 1).trans hn,?_⟩ |
| calc |
| _ ≤ (ball K S r).ncard := Set.ncard_le_ncard |
| (Set.image_mono ((hin n).trans (Set.pow_subset_pow_right h1 hnr))) (hg.1 r) |
| _ ≤ C*r^d := hbound |
| _ ≤ C*((2*(L+1))*n)^d := Nat.mul_le_mul_left C (Nat.pow_le_pow_left hrn d) |
| _ = _ := by rw [mul_pow]; ring |
|
|
| lemma HasDegree.zero_finiteIndex {K : Subgroup A} {S : Set A} |
| (h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤) |
| (hg : HasDegree K S 0) : K.FiniteIndex := by |
| obtain ⟨C,hC⟩ := hg.2 |
| obtain ⟨n,hn,hb⟩ := hC C |
| apply CosetGrowth.finiteIndex_of_small_ball S K h1 hi hgen n (hg.1 n) |
| exact (by simpa using hb : (ball K S n).ncard≤C).trans hn |
|
|
| end CriticalPercolation.RelativeGrowth |
| end |
| end |
|
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| end OAI |
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