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2fca478 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import OAI.Geometry.PinnedDistances.NormalizedGraphs
namespace OAI
universe u1 u2 u3 u4
section
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators
namespace WeakPinned.Bounded
open Hierarchy Global Variance NumberField
lemma bounded_denominator_rate (x w : ℕ → ℝ) (C : ℝ)
(hx : ∀ n,0≤x n) (hw : ∀ n,0≤w n) (hC : ∀ n,w n≤C)
(ht : Tendsto (fun n => x n/(w n+1)) atTop (𝓝 0)) :
Tendsto x atTop (𝓝 0) := by
apply squeeze_zero hx _ (by simpa only [mul_zero] using tendsto_const_nhds.mul ht (a:=C+1))
intro n
have hp : 0<w n+1 := by linarith [hw n]
calc
x n = (w n+1)*(x n/(w n+1)) := by field_simp
_ ≤ (C+1)*(x n/(w n+1)) :=
mul_le_mul_of_nonneg_right (by linarith [hC n]) (div_nonneg (hx n) hp.le)
variable {P : Type u1} [Fintype P] [DecidableEq P]
variable {K : Type u2} [Field K] [NumberField K]
lemma massEval_add (q r : Finset P → ℝ) (d d' : Depth P) :
massEval (fun C => q C+r C) d d'=massEval q d d'+massEval r d d' := by
simp only [massEval,weightedIntegral,mul_add,Finset.sum_add_distrib]
ring
omit [NumberField K] in
lemma archMean_mass_add (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
(hn : 2≤Fintype.card P) (q r : Finset P → ℝ) (v : InfinitePlace K) :
archMean Z₁ Z₂ (massEval (fun C => q C+r C)) v=
archMean Z₁ Z₂ (massEval q) v+archMean Z₁ Z₂ (massEval r) v := by
unfold archMean
simp only [massEval_add]
exact integral_add (integrable_massEval Z₁ Z₂ h₁ h₂ hn q v)
(integrable_massEval Z₁ Z₂ h₁ h₂ hn r v)
omit [NumberField K] in
lemma archMean_mass_swap (Z₁ Z₂ : P → K) (q : Finset P → ℝ) (v : InfinitePlace K) :
archMean Z₂ Z₁ (massEval q) v=archMean Z₁ Z₂ (massEval q) v := by
unfold archMean
congr 1
funext ω
exact add_comm _ _
omit [NumberField K] in
lemma archMean_scale_swap (Z₁ Z₂ : P → K) (v : InfinitePlace K) :
archMean Z₂ Z₁ scaleEval v=archMean Z₁ Z₂ scaleEval v := by
unfold archMean
congr 1
funext ω
simp only [scaleEval,massEval,potentialEval]
congr 1
· exact add_comm _ _
· apply Finset.sum_congr rfl
intro x _
rw [add_comm ((arch Z₂ v ω).potential x)]
omit [DecidableEq P] in
lemma weightedIntegral_add (m q r : Finset P → ℝ) :
weightedIntegral m (fun C => q C+r C)=weightedIntegral m q+weightedIntegral m r := by
simp only [weightedIntegral,mul_add,Finset.sum_add_distrib]
variable {V : ℕ → Type u3} {L : ℕ → Type u4} [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)]
[∀ n,Field (L n)] [∀ n,NumberField (L n)]
lemma bounded_place_selection (Z₁ Z₂ : ∀ n,V n → L n)
(h₁ : ∀ n,Function.Injective (Z₁ n)) (h₂ : ∀ n,Function.Injective (Z₂ n))
(hn : ∀ n,2≤Fintype.card (V n)) (q r : ∀ n,Finset (V n) → ℝ)
(hq : ∀ n C,0≤q n C) (hr : ∀ n C,0≤r n C)
(C : ℝ) (hC : 0≤C)
(hscale : ∀ n,overlapScale (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)+
potentialScale (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)≤C)
(hqt : Tendsto (fun n => weightedIntegral (totalMass (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)) (q n)) atTop (𝓝 0))
(hrt : Tendsto (fun n => weightedIntegral (totalMass (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)) (r n)) atTop (𝓝 0)) :
∃ v : ∀ n,InfinitePlace (L n),
(∀ n,archMean (Z₁ n) (Z₂ n) scaleEval (v n)≤2*C+1) ∧
Tendsto (fun n => archMean (Z₁ n) (Z₂ n) (massEval (q n)) (v n)) atTop (𝓝 0) ∧
Tendsto (fun n => archMean (Z₁ n) (Z₂ n) (massEval (r n)) (v n)) atTop (𝓝 0) := by
let m := fun n => totalMass (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)
let ε := fun n => weightedIntegral (m n) (q n)+weightedIntegral (m n) (r n)+1/((n:ℝ)+1)
have hq0 (n) : 0≤weightedIntegral (m n) (q n) :=
Finset.sum_nonneg (fun D _ => mul_nonneg (totalMass_nonneg _ _ _ _ _) (hq n D))
have hr0 (n) : 0≤weightedIntegral (m n) (r n) :=
Finset.sum_nonneg (fun D _ => mul_nonneg (totalMass_nonneg _ _ _ _ _) (hr n D))
have he (n) : 0<ε n := by
have hh : 0<1/((n:ℝ)+1) := by positivity
dsimp only [ε]
linarith [hq0 n,hr0 n]
have het : Tendsto ε atTop (𝓝 0) := by
have hi : Tendsto (fun n : ℕ => 1/((n:ℝ)+1)) atTop (𝓝 0) := by
simpa only [one_div,Function.comp_def] using tendsto_inv_atTop_zero.comp (tendsto_atTop_add_const_right _ _ tendsto_natCast_atTop_atTop)
simpa only [add_zero] using (hqt.add hrt).add hi
have hs (n) := select_arch (Z₁ n) (Z₂ n) (h₁ n) (h₂ n) (hn n)
(fun D => q n D+r n D) (fun D => add_nonneg (hq n D) (hr n D)) C (ε n) hC (he n) (hscale n)
(by rw [weightedIntegral_add]; dsimp [ε,m]; linarith [show 0≤1/((n:ℝ)+1) by positivity])
choose v hvscale hv using hs
have hvq (n) : 0≤archMean (Z₁ n) (Z₂ n) (massEval (q n)) (v n) :=
archMean_nonneg _ _ _ (massEval_nonneg _ (hq n)) _
have hvr (n) : 0≤archMean (Z₁ n) (Z₂ n) (massEval (r n)) (v n) :=
archMean_nonneg _ _ _ (massEval_nonneg _ (hr n)) _
have hsum (n) := hv n
simp only [archMean_mass_add _ _ (h₁ _) (h₂ _) (hn _)] at hsum
refine ⟨v,hvscale,?_,?_⟩
· apply squeeze_zero hvq _ (by simpa only [mul_zero] using het.const_mul 2)
intro n
linarith [hsum n,hvr n]
· apply squeeze_zero hvr _ (by simpa only [mul_zero] using het.const_mul 2)
intro n
linarith [hsum n,hvq n]
lemma power_fraction_rate (n : ℕ → ℝ) (hn : ∀ j,0<n j)
(ht : Tendsto n atTop atTop) (c : ℝ) (hc : 0<c) :
Tendsto (fun j => (n j)^(1-c)/n j) atTop (𝓝 0) := by
have hh := (tendsto_rpow_neg_atTop hc).comp ht
convert hh using 1
funext j
rw [show 1-c=1+(-c) by ring,Real.rpow_add (hn j),Real.rpow_one]
simp only [Function.comp_apply]
exact mul_div_cancel_left₀ _ (hn j).ne'
end WeakPinned.Bounded
end
end
end OAI
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