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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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
end OAI
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