| import Mathlib |
| import OAI.Probability.CriticalPercolation.Growth.CosetTransfer |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical Pointwise |
|
|
| namespace CriticalPercolation.RelativeGrowth |
| variable {A : Type*} [Group A] |
|
|
| lemma integer_kernel_coset_packing (K : Subgroup A) (S : Set A) (h1 : (1:A)∈S) |
| (f : A →* Multiplicative ℤ) (hKf : K≤f.ker) (a : A) (ha : f a=Multiplicative.ofAdd 1) |
| (T : Set f.ker) (L M : ℕ) (hT : f.ker.subtype '' T⊆S^L) (haS : a∈S^M) |
| (n : ℕ) (hfin : (ball K S ((L+M+1)*n)).Finite) : |
| n*(ball (K.subgroupOf f.ker) T n).ncard≤(ball K S ((L+M+1)*n)).ncard := by |
| let Q := ball (K.subgroupOf f.ker) T n |
| have hh : ∀ q∈Q, ∃ b : f.ker, b∈T^n ∧ (QuotientGroup.mk b : f.ker ⧸ K.subgroupOf f.ker)=q := fun q hq => hq |
| choose b hb he using hh |
| let phi : Fin n × Q → A ⧸ K := fun p => QuotientGroup.mk (a^p.1.val*(b p.2.val p.2.property).val) |
| have hin (p : Fin n × Q) : phi p∈ball K S ((L+M+1)*n) := by |
| refine ⟨a^p.1.val*(b p.2.val p.2.property).val,?_,rfl⟩ |
| have ha' : a^p.1.val∈S^(M*p.1.val) := by |
| rw [pow_mul] |
| exact Set.pow_mem_pow haS |
| have hb' : (b p.2.val p.2.property).val∈S^(L*n) := by |
| rw [pow_mul] |
| apply Set.pow_subset_pow_left hT |
| rw [←Set.image_pow] |
| exact ⟨b p.2.val p.2.property,hb p.2.val p.2.property,rfl⟩ |
| apply Set.pow_subset_pow_right h1 (show M*p.1.val+L*n≤(L+M+1)*n by nlinarith [p.1.isLt]) |
| rw [pow_add] |
| exact Set.mul_mem_mul ha' hb' |
| have hphi : Injective phi := by |
| rintro ⟨i,q⟩ ⟨j,r⟩ hij |
| have hm := QuotientGroup.eq.mp hij |
| have hf' := MonoidHom.mem_ker.mp (hKf hm) |
| simp only [map_mul,map_inv,map_pow,ha,MonoidHom.mem_ker.mp (b q.val q.property).property, |
| MonoidHom.mem_ker.mp (b r.val r.property).property,mul_one] at hf' |
| have hi : i=j := by |
| apply Fin.ext |
| have hz := congrArg Multiplicative.toAdd hf' |
| simp at hz |
| omega |
| subst j |
| apply congrArg (fun s : Q => (i,s)) |
| apply Subtype.ext |
| rw [←he q.val q.property,←he r.val r.property] |
| apply QuotientGroup.eq.mpr |
| change ((b q.val q.property).val)⁻¹*(b r.val r.property).val∈K |
| simpa only [mul_inv_rev,mul_assoc,inv_mul_cancel_left] using hm |
| let e : Fin n × Q ↪ ball K S ((L+M+1)*n) := ⟨fun p => ⟨phi p,hin p⟩,fun p q h => hphi (congrArg Subtype.val h)⟩ |
| let : Finite (ball K S ((L+M+1)*n)) := hfin.to_subtype |
| have hc := Nat.card_le_card_of_injective e e.injective |
| simpa only [Nat.card_prod,Nat.card_fin,Nat.card_coe_set_eq,Q] using hc |
|
|
| lemma HasDegree.integer_kernel_descent {K : Subgroup A} {S : Set A} {d : ℕ} |
| (h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤) |
| (hg : HasDegree K S (d+1)) (f : A →* Multiplicative ℤ) (hf : Surjective f) (hKf : K≤f.ker) |
| (T : Set f.ker) (hT : Bounded S (f.ker.subtype '' T)) : |
| HasDegree (K.subgroupOf f.ker) T d := by |
| have hTf := (hg.subgroup_bounded h1 f.ker T hT).1 |
| obtain ⟨L,hL⟩ := hT |
| obtain ⟨a,ha⟩ := hf (Multiplicative.ofAdd 1) |
| obtain ⟨M,hM⟩ := bounded_singleton S h1 hi hgen a |
| have haS := hM (Set.mem_singleton a) |
| obtain ⟨C,hC⟩ := hg.2 |
| let D := L+M+1 |
| have hD : 0<D := by dsimp [D]; omega |
| refine ⟨hTf,C*(2*D)^(d+1),?_⟩ |
| intro R |
| obtain ⟨r,hr,hbound⟩ := hC (D*max R 1) |
| let n := r/D |
| have hnr : D*n≤r := by simpa only [n,Nat.mul_comm] using Nat.div_mul_le_self r D |
| have hn : max R 1≤n := (Nat.le_div_iff_mul_le hD).mpr (by simpa [Nat.mul_comm] using hr) |
| have hn1 : 1≤n := (le_max_right R 1).trans hn |
| have hrn : r≤(2*D)*n := by |
| have hlt := Nat.lt_mul_div_succ r hD |
| change r<D*(n+1) at hlt |
| nlinarith |
| refine ⟨n,(le_max_left R 1).trans hn,?_⟩ |
| have hb : n*(ball (K.subgroupOf f.ker) T n).ncard≤n*(C*(2*D)^(d+1)*n^d) := by |
| calc |
| _ ≤ (ball K S (D*n)).ncard := integer_kernel_coset_packing K S h1 f hKf a ha T L M hL haS n (hg.1 _) |
| _ ≤ (ball K S r).ncard := Set.ncard_le_ncard (ball_mono K S h1 hnr) (hg.1 _) |
| _ ≤ C*r^(d+1) := hbound |
| _ ≤ C*((2*D)*n)^(d+1) := Nat.mul_le_mul_left C (Nat.pow_le_pow_left hrn (d+1)) |
| _ = _ := by rw [mul_pow,pow_succ n]; ring |
| exact Nat.le_of_mul_le_mul_left hb (by omega : 0<n) |
|
|
| end CriticalPercolation.RelativeGrowth |
| end |
| end |
|
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| end OAI |
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