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8769036 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 | import OAI.NumberTheory.TwoPoint.Bounds.Dilation
import OAI.NumberTheory.TwoPoint.Bounds.AffineTransfer
import OAI.NumberTheory.TwoPoint.Statements
namespace OAI
open Filter
open scoped BigOperators
namespace TwoPointCorrelations
def ProgressionCorrectedElliott : Prop :=
∀ f g : ℕ → ℂ, Multiplicative f → Multiplicative g →
OneBounded f → OneBounded g →
(UniformlyNonpretentious f ∨ UniformlyNonpretentious g) →
∀ h l b : ℕ, 0 < h → 0 < l →
Tendsto (fun N : ℕ => residuePrefix (fun n => f n * g (n + h)) l b N / (N : ℂ))
atTop (nhds 0)
theorem Multiplicative.one_eq_zero_or_one {f : ℕ → ℂ} (hf : Multiplicative f) :
f 1 = 0 ∨ f 1 = 1 := by
have heq := hf 1 1 (by decide) (by decide) (by decide)
have hprod : f 1 * (f 1 - 1) = 0 := by
calc
f 1 * (f 1 - 1) = f 1 * f 1 - f 1 := by ring
_ = 0 := by simpa using congrArg (fun z : ℂ => z - f 1) heq.symm
rcases mul_eq_zero.mp hprod with h | h
· exact Or.inl h
· exact Or.inr (sub_eq_zero.mp h)
theorem Multiplicative.zero_of_one_eq_zero {f : ℕ → ℂ} (hf : Multiplicative f)
(h0 : f 1 = 0) {n : ℕ} (hn : 0 < n) : f n = 0 := by
simpa [h0] using hf 1 n (by decide) hn (Nat.coprime_one_left n)
lemma residuePrefix_congr {u v : ℕ → ℂ} (h : ∀ n, 0 < n → u n = v n)
(l b N : ℕ) : residuePrefix u l b N = residuePrefix v l b N := by
unfold residuePrefix
apply Finset.sum_congr rfl
intro n hn
rw [h n (Finset.mem_Ioc.mp hn).1]
lemma residuePrefix_sum {ι : Type*} (s : Finset ι) (u : ι → ℕ → ℂ) (l b N : ℕ) :
residuePrefix (fun n => ∑ i ∈ s, u i n) l b N =
∑ i ∈ s, residuePrefix (u i) l b N := by
unfold residuePrefix
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro n hn
split_ifs <;> simp
lemma residuePrefix_const_mul (c : ℂ) (u : ℕ → ℂ) (l b N : ℕ) :
residuePrefix (fun n => c * u n) l b N = c * residuePrefix u l b N := by
unfold residuePrefix
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro n hn
split_ifs <;> simp
/-- Finite signed local expansions preserve an established progression limit. -/
theorem finite_expansion_progression_zero {ι κ : Type*}
(s : Finset ι) (t : Finset κ) (c : ι → ℂ) (d : κ → ℂ)
(F : ι → ℕ → ℂ) (G : κ → ℕ → ℂ) (u v : ℕ → ℂ)
(hu : ∀ n, 0 < n → u n = ∑ i ∈ s, c i * F i n)
(hv : ∀ n, 0 < n → v n = ∑ j ∈ t, d j * G j n)
(h l b : ℕ)
(hz : ∀ i ∈ s, ∀ j ∈ t,
Tendsto (fun N : ℕ => residuePrefix (fun n => F i n * G j (n + h)) l b N /
(N : ℂ)) atTop (nhds 0)) :
Tendsto (fun N : ℕ => residuePrefix (fun n => u n * v (n + h)) l b N /
(N : ℂ)) atTop (nhds 0) := by
have heq (N : ℕ) : residuePrefix (fun n => u n * v (n + h)) l b N / (N : ℂ) =
∑ i ∈ s, ∑ j ∈ t, (c i * d j) *
(residuePrefix (fun n => F i n * G j (n + h)) l b N / (N : ℂ)) := by
rw [residuePrefix_congr (v := fun n => ∑ i ∈ s, ∑ j ∈ t,
(c i * d j) * (F i n * G j (n + h))) (by
intro n hn
rw [hu n hn, hv (n + h) (by omega)]
rw [Finset.sum_mul]
simp_rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i hi
apply Finset.sum_congr rfl
intro j hj
ring)]
simp only [residuePrefix_sum, residuePrefix_const_mul, Finset.sum_div]
apply Finset.sum_congr rfl
intro i hi
apply Finset.sum_congr rfl
intro j hj
ring
simp_rw [heq]
have hout := tendsto_finsetSum s (fun i hi =>
tendsto_finsetSum t (fun j hj =>
(tendsto_const_nhds (x := c i * d j)).mul (hz i hi j hj)))
simpa using hout
theorem ProgressionCorrectedElliott.dilations
(hprog : ProgressionCorrectedElliott) (f g : ℕ → ℂ)
(hf : Multiplicative f) (hg : Multiplicative g)
(hf1 : f 1 = 1) (hg1 : g 1 = 1)
(hfb : OneBounded f) (hgb : OneBounded g)
(hnp : UniformlyNonpretentious f ∨ UniformlyNonpretentious g)
(a a' h l b : ℕ) (ha : 0 < a) (ha' : 0 < a') (hh : 0 < h) (hl : 0 < l) :
Tendsto (fun N : ℕ =>
residuePrefix (fun n => dilate a f n * dilate a' g (n + h)) l b N / (N : ℂ))
atTop (nhds 0) := by
apply finite_expansion_progression_zero a.primeFactors.powerset a'.primeFactors.powerset
(fun E => (-1 : ℂ) ^ E.card) (fun E => (-1 : ℂ) ^ E.card)
(dilationComponent a f) (dilationComponent a' g)
(dilate a f) (dilate a' g)
(fun n hn => dilate_eq_sum_components a f ha hf hf1 hn)
(fun n hn => dilate_eq_sum_components a' g ha' hg hg1 hn)
intro E _ E' _
apply hprog _ _ (dilationComponent_multiplicative a f E)
(dilationComponent_multiplicative a' g E')
(dilationComponent_oneBounded _ _ _ hfb) (dilationComponent_oneBounded _ _ _ hgb)
_ h l b hh hl
exact hnp.imp (dilationComponent_nonpretentious a f E hfb)
(dilationComponent_nonpretentious a' g E' hgb)
private theorem affine_ordered_zero (hprog : ProgressionCorrectedElliott)
(f g : ℕ → ℂ) (hf : Multiplicative f) (hg : Multiplicative g)
(hf1 : f 1 = 1) (hg1 : g 1 = 1) (hfb : OneBounded f) (hgb : OneBounded g)
(hnp : UniformlyNonpretentious f ∨ UniformlyNonpretentious g)
(a₁ a₂ b₁ b₂ : ℕ) (ha₁ : 0 < a₁) (ha₂ : 0 < a₂)
(horder : a₂ * b₁ < a₁ * b₂) :
Tendsto (fun N : ℕ => affineSum f g a₁ a₂ b₁ b₂ N / (N : ℂ)) atTop (nhds 0) := by
have hp := hprog.dilations f g hf hg hf1 hg1 hfb hgb hnp a₂ a₁
(a₁ * b₂ - a₂ * b₁) (a₁ * a₂) (a₂ * b₁) ha₂ ha₁
(Nat.sub_pos_of_lt horder) (Nat.mul_pos ha₁ ha₂)
have ht := affine_mean_tendsto_zero
(fun m => dilate a₂ f m * dilate a₁ g (m + (a₁ * b₂ - a₂ * b₁)))
(a₁ * a₂) (a₂ * b₁) (Nat.mul_pos ha₁ ha₂) hp
apply ht.congr'
apply Filter.Eventually.of_forall
intro N
dsimp only
congr 1
unfold affineSum
apply Finset.sum_congr rfl
intro n hn
have hl : a₁ * a₂ * n + a₂ * b₁ = a₂ * (a₁ * n + b₁) := by ring
have hr : a₁ * a₂ * n + a₂ * b₁ + (a₁ * b₂ - a₂ * b₁) =
a₁ * (a₂ * n + b₂) := by
calc
_ = a₁ * a₂ * n + a₁ * b₂ := by omega
_ = _ := by ring
rw [hr, hl]
simp [dilate, ha₁.ne', ha₂.ne']
/-- The manuscript's affine reduction, conditional here on its internal
progression theorem. This is not the published-input conditional main result. -/
theorem ProgressionCorrectedElliott.affine (hprog : ProgressionCorrectedElliott) :
AffineCorrectedElliott := by
intro f g hf hg hfb hgb hnp a₁ a₂ b₁ b₂ ha₁ ha₂ hneq
rcases hf.one_eq_zero_or_one with hf0 | hf1
· have hz (N : ℕ) : affineSum f g a₁ a₂ b₁ b₂ N = 0 := by
unfold affineSum
apply Finset.sum_eq_zero
intro n hn
have hnpos := (Finset.mem_Icc.mp hn).1
rw [hf.zero_of_one_eq_zero hf0 (by nlinarith : 0 < a₁ * n + b₁), zero_mul]
simpa only [hz, zero_div] using (tendsto_const_nhds (x := (0 : ℂ)) (f := atTop))
rcases hg.one_eq_zero_or_one with hg0 | hg1
· have hz (N : ℕ) : affineSum f g a₁ a₂ b₁ b₂ N = 0 := by
unfold affineSum
apply Finset.sum_eq_zero
intro n hn
have hnpos := (Finset.mem_Icc.mp hn).1
rw [hg.zero_of_one_eq_zero hg0 (by nlinarith : 0 < a₂ * n + b₂), mul_zero]
simpa only [hz, zero_div] using (tendsto_const_nhds (x := (0 : ℂ)) (f := atTop))
rcases lt_or_gt_of_ne hneq with hlt | hgt
· have h := affine_ordered_zero hprog g f hg hf hg1 hf1 hgb hfb hnp.symm
a₂ a₁ b₂ b₁ ha₂ ha₁ hlt
simpa only [affineSum_swap f g] using h
· exact affine_ordered_zero hprog f g hf hg hf1 hg1 hfb hgb hnp
a₁ a₂ b₁ b₂ ha₁ ha₂ hgt
end TwoPointCorrelations
end OAI
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