| import OAI.NumberTheory.Jacobsthal.Estimates.HarmonicConditioning |
| import OAI.NumberTheory.Jacobsthal.Estimates.SourceInitialDiscard |
|
|
| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| namespace ErdosInverseCells |
| attribute [local instance] Classical.decEq |
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| theorem weighted_to_count_density {α : Type*} (A B : Finset α) (hB : B ⊆ A) (hA : 0 < A.card) |
| (w : α → ℝ) (G rho : ℝ) (hG : 0 < G) (hTotal : 0 < ∑ a ∈ A,w a) |
| (hspread : ∀ a ∈ A,∀ b ∈ A,w a ≤ G*w b) |
| (hweighted : rho*(∑ a ∈ A,w a) ≤ ∑ b ∈ B,w b) : |
| (rho/G)*(A.card : ℝ) ≤ (B.card : ℝ) := by |
| have hspreadSum := subset_weight_spread A B hB hA w G hspread |
| have hcoef : rho ≤ (B.card : ℝ)*G/(A.card : ℝ) := |
| le_of_mul_le_mul_right (hweighted.trans hspreadSum) hTotal |
| have hApos : (0 : ℝ) < A.card := by exact_mod_cast hA |
| have hmul := (le_div_iff₀ hApos).mp hcoef |
| calc |
| _ = (rho*(A.card : ℝ))/G := by ring |
| _ ≤ (B.card : ℝ) := (div_le_iff₀ hG).mpr hmul |
|
|
| end ErdosInverseCells |
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| end |
|
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| section |
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|
| namespace ErdosInverseCells |
| open ErdosInverseSampling |
| attribute [local instance] Classical.decEq |
| attribute [local instance] Classical.propDecidable |
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|
| theorem triple_weight_spread (P Q U : Finset ℕ) (xi : ℝ) (hxi : 0 ≤ xi) (hxi1 : xi ≤ 1) |
| (hP : ∀ p ∈ P,∀ r ∈ P,(p : ℝ)⁻¹ ≤ (1+xi)*(r : ℝ)⁻¹) |
| (hQ : ∀ q ∈ Q,∀ r ∈ Q,(q : ℝ)⁻¹ ≤ (1+xi)*(r : ℝ)⁻¹) |
| (hU : ∀ u ∈ U,∀ r ∈ U,(u : ℝ)⁻¹ ≤ (1+xi)*(r : ℝ)⁻¹) : |
| ∀ x ∈ tripleSpace P Q U,∀ y ∈ tripleSpace P Q U,tripleWeight x ≤ 8*tripleWeight y := by |
| intro x hx y hy |
| have hx' : x.1.1 ∈ P ∧ x.1.2 ∈ Q ∧ x.2 ∈ U := by |
| simpa only [tripleSpace,Finset.mem_product,and_assoc] using hx |
| have hy' : y.1.1 ∈ P ∧ y.1.2 ∈ Q ∧ y.2 ∈ U := by |
| simpa only [tripleSpace,Finset.mem_product,and_assoc] using hy |
| have h1 := hP _ hx'.1 _ hy'.1 |
| have h2 := hQ _ hx'.2.1 _ hy'.2.1 |
| have h3 := hU _ hx'.2.2 _ hy'.2.2 |
| have hc : (1+xi)^3 ≤ (8 : ℝ) := by |
| have hh := pow_le_pow_left₀ (by linarith : 0 ≤ 1+xi) (by linarith : 1+xi ≤ 2) 3 |
| norm_num at hh |
| exact hh |
| calc |
| _ ≤ ((1+xi)*(y.1.1 : ℝ)⁻¹)*((1+xi)*(y.1.2 : ℝ)⁻¹)*((1+xi)*(y.2 : ℝ)⁻¹) := |
| mul_le_mul (mul_le_mul h1 h2 (by positivity) (by positivity)) h3 (by positivity) (by positivity) |
| _ = (1+xi)^3*tripleWeight y := by unfold tripleWeight;ring |
| _ ≤ _ := mul_le_mul_of_nonneg_right hc (tripleWeight_nonneg y) |
|
|
| theorem triple_weighted_to_count (P Q U : Finset ℕ) (xi sigma : ℝ) (hxi : 0 ≤ xi) (hxi1 : xi ≤ 1) |
| (witness : (ℕ × ℕ) → ℕ → Prop) |
| (hPpos : 0 < harmonicMass P) (hQpos : 0 < harmonicMass Q) (hUpos : 0 < harmonicMass U) |
| (hP : ∀ p ∈ P,∀ r ∈ P,(p : ℝ)⁻¹ ≤ (1+xi)*(r : ℝ)⁻¹) |
| (hQ : ∀ q ∈ Q,∀ r ∈ Q,(q : ℝ)⁻¹ ≤ (1+xi)*(r : ℝ)⁻¹) |
| (hU : ∀ u ∈ U,∀ r ∈ U,(u : ℝ)⁻¹ ≤ (1+xi)*(r : ℝ)⁻¹) |
| (hweighted : (sigma/2)*(harmonicMass P*harmonicMass Q*harmonicMass U) ≤ weightedWitness P Q U witness) : |
| (sigma/16)*((P.product Q).card : ℝ)*(U.card : ℝ) ≤ |
| ((witnessPairs (P.product Q) U witness).card : ℝ) := by |
| have hTotal : 0 < ∑ x ∈ tripleSpace P Q U,tripleWeight x := by |
| rw [triple_weight_sum] |
| exact mul_pos (mul_pos hPpos hQpos) hUpos |
| have hA : 0 < (tripleSpace P Q U).card := by |
| by_contra hn |
| have he : tripleSpace P Q U = ∅ := Finset.card_eq_zero.mp (by omega) |
| rw [he,Finset.sum_empty] at hTotal |
| linarith |
| have hsub : witnessPairs (P.product Q) U witness ⊆ tripleSpace P Q U := Finset.filter_subset _ _ |
| have hh := weighted_to_count_density (tripleSpace P Q U) (witnessPairs (P.product Q) U witness) |
| hsub hA tripleWeight 8 (sigma/2) (by norm_num) hTotal (triple_weight_spread P Q U xi hxi hxi1 hP hQ hU) |
| (by simpa only [triple_weight_sum,weightedWitness] using hweighted) |
| simpa only [tripleSpace,Finset.product_eq_sprod,Finset.card_product,Nat.cast_mul,div_div,show (2 : ℝ)*8 = 16 by norm_num,mul_assoc] using hh |
|
|
| end ErdosInverseCells |
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|
| end |
|
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| end Erdos970 |
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| end OAI |
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|
