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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Harmonic.GeneralizedGramSpectrum
namespace OAI
section
noncomputable section
open Set Function Module
open scoped Classical BigOperators Matrix
namespace CriticalPercolation.HarmonicAnalysis
variable {ι : Type*} [Fintype ι]
lemma many_small_factors (μ : ι → ℝ) (hμ : ∀ i, 1 ≤ μ i)
(C : ℝ) (hC : 1 < C) (hprod : (∏ i, μ i) ≤ C^(Fintype.card ι)) :
Fintype.card ι ≤ 2*(Finset.univ.filter (fun i => μ i ≤ C^2)).card := by
let I := Finset.univ.filter (fun i => μ i ≤ C^2)
let J := Finset.univ.filter (fun i => C^2 < μ i)
have hj (i : ι) : (if i∈J then C^2 else 1) ≤ μ i := by
split_ifs with hi
· exact (Finset.mem_filter.mp hi).2.le
· exact hμ i
have hp := Finset.prod_le_prod₀ (fun i (_ : i∈Finset.univ) => by
split_ifs <;> positivity) (fun i (_ : i∈Finset.univ) => hj i)
have he : (∏ i : ι, if i∈J then C^2 else 1)=(C^2)^J.card := by
rw [←Finset.prod_filter]
simp
rw [he,←pow_mul] at hp
have hcard : 2*J.card ≤ Fintype.card ι := (pow_le_pow_iff_right₀ hC).mp (hp.trans hprod)
have hIJ : I.card+J.card=Fintype.card ι := by
have hh := Finset.card_filter_add_card_filter_not
(s := (Finset.univ : Finset ι)) (p := fun i => μ i ≤ C^2)
simpa only [not_le,Finset.card_univ] using hh
change Fintype.card ι ≤ 2*I.card
omega
omit [Fintype ι] in
lemma basis_support_span {E : Type*} [AddCommGroup E] [Module ℝ E] (b : Basis ι ℝ E) (I : Finset ι)
{x : E} (hx : x∈Submodule.span ℝ (Set.range (fun i:I => b i.val))) :
∀ i∉I, b.repr x i=0 := by
induction hx using Submodule.span_induction with
| mem x hx =>
obtain ⟨j,rfl⟩ := hx
intro i hi
have hj : j.val≠i := by intro he; exact hi (he ▸ j.property)
simp [hj]
| zero => simp
| add x y hx hy ix iy =>
intro i hi
simp only [map_add,Finsupp.add_apply,ix i hi,iy i hi,add_zero]
| smul a x hx ix =>
intro i hi
simp only [map_smul,Finsupp.smul_apply,smul_eq_mul,ix i hi,mul_zero]
theorem exists_low_growth_subspace (A B : Matrix ι ι ℝ) (hA : A.PosDef)
(hB : B.IsHermitian) (hmono : ∀ x, matrixQuadratic A x ≤ matrixQuadratic B x)
(C : ℝ) (hC : 1 < C) (hdet : B.det ≤ C^(Fintype.card ι)*A.det) :
∃ V : Submodule ℝ (ι → ℝ),
Fintype.card ι ≤ 2*Module.finrank ℝ V ∧
∀ x∈V, matrixQuadratic B x ≤ C^2*matrixQuadratic A x := by
obtain ⟨μ,b,hμ,hd,hqa,hqb⟩ := generalized_gram_spectrum A B hA hB hmono
have hp : (∏ i, μ i) ≤ C^(Fintype.card ι) := by
rw [hd,mul_comm (C^(Fintype.card ι))] at hdet
exact (mul_le_mul_iff_right₀ hA.det_pos).mp (by simpa only [mul_comm A.det] using hdet)
let I := Finset.univ.filter (fun i => μ i ≤ C^2)
let V := Submodule.span ℝ (Set.range (fun i : I => b i.val))
have hind : LinearIndependent ℝ (fun i : I => b i.val) :=
b.linearIndependent.comp Subtype.val Subtype.val_injective
have hvdim : Module.finrank ℝ V=I.card := by
exact (finrank_span_eq_card hind).trans (Fintype.card_coe I)
refine ⟨V,?_,?_⟩
· rw [hvdim]
simpa only [Fintype.card_fin] using many_small_factors μ hμ C hC (by simpa using hp)
· intro x hx
rw [hqb,hqa,Finset.mul_sum]
have hs := basis_support_span b I hx
apply Finset.sum_le_sum
intro i hi
by_cases hiI : i∈I
· exact mul_le_mul_of_nonneg_right (Finset.mem_filter.mp hiI).2 (sq_nonneg _)
· rw [hs i hiI,pow_two,zero_mul,mul_zero,mul_zero]
end CriticalPercolation.HarmonicAnalysis
end
end
end OAI
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