| 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 |
|
|
