| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators |
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| namespace Problem335 |
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| / |
| its total absolute discrepancy. This is the algebraic identity used when |
| ordering the factors in the product-rank argument. -/ |
| theorem sum_positive_eq_half_sum_abs {ι : Type*} (s : Finset ι) (δ : ι → ℝ) |
| (hδ : ∑ j ∈ s, δ j = 0) : |
| ∑ j ∈ s.filter (fun j => 0 < δ j), δ j = |
| (∑ j ∈ s, |δ j|) / 2 := by |
| classical |
| have habs : ∀ j, |δ j| = 2 * (if 0 < δ j then δ j else 0) - δ j := by |
| intro j |
| by_cases h : 0 < δ j |
| · rw [ite_eq_left h, abs_of_pos h] |
| ring |
| · rw [ite_eq_right h, abs_of_nonpos (le_of_not_gt h)] |
| ring |
| have hs : (∑ j ∈ s, |δ j|) = |
| 2 * (∑ j ∈ s.filter (fun j => 0 < δ j), δ j) := by |
| simp_rw [habs] |
| rw [Finset.sum_sub_distrib, ← Finset.mul_sum, hδ, sub_zero, |
| Finset.sum_filter] |
| linarith |
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| / |
| theorem sum_degree_discrepancy_eq_zero {ι : Type*} (s : Finset ι) |
| (e i : ι → ℝ) (lam n k : ℝ) |
| (he : ∑ j ∈ s, e j = n) (hi : ∑ j ∈ s, i j = k) |
| (hlam : lam * n = k) : |
| ∑ j ∈ s, (i j - lam * e j) = 0 := by |
| rw [Finset.sum_sub_distrib, ← Finset.mul_sum, he, hi, hlam, sub_self] |
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| / |
| precisely the dimension-saving exponent in terms of absolute discrepancies. -/ |
| theorem positive_bidegree_identity {ι : Type*} (s : Finset ι) |
| (e i : ι → ℝ) (ρ lam : ℝ) |
| (hlam : (1 + ρ) * lam = ρ) |
| (hδ : ∑ j ∈ s, (i j - lam * e j) = 0) : |
| (∑ j ∈ s.filter (fun j => 0 < i j - lam * e j), i j) - |
| ρ * (∑ j ∈ s.filter (fun j => 0 < i j - lam * e j), (e j - i j)) = |
| (1 + ρ) / 2 * (∑ j ∈ s, |i j - lam * e j|) := by |
| classical |
| have hpoint : ∀ j, i j - ρ * (e j - i j) = |
| (1 + ρ) * (i j - lam * e j) := by |
| intro j |
| nlinarith [congrArg (fun x : ℝ => x * e j) hlam] |
| rw [Finset.mul_sum, ← Finset.sum_sub_distrib] |
| simp_rw [hpoint] |
| rw [← Finset.mul_sum, sum_positive_eq_half_sum_abs s (fun j => i j - lam * e j) hδ] |
| ring |
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| / |
| individual discrepancy weights. -/ |
| theorem positive_bidegree_rpow {ι : Type*} (s : Finset ι) |
| (e i : ι → ℝ) (ρ lam α : ℝ) (hα : 0 < α) |
| (hlam : (1 + ρ) * lam = ρ) |
| (hδ : ∑ j ∈ s, (i j - lam * e j) = 0) : |
| α ^ ((∑ j ∈ s.filter (fun j => 0 < i j - lam * e j), i j) - |
| ρ * (∑ j ∈ s.filter (fun j => 0 < i j - lam * e j), (e j - i j))) = |
| ∏ j ∈ s, (α ^ ((1 + ρ) / 2)) ^ |i j - lam * e j| := by |
| rw [positive_bidegree_identity s e i ρ lam hlam hδ, |
| Real.rpow_mul hα.le, |
| Real.rpow_sum_of_pos (Real.rpow_pos_of_pos hα _) _ s] |
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| / |
| finite sums is valid for nonnegative weights. -/ |
| theorem constrained_sum_prod_le_prod_sum {ι : Type*} [Fintype ι] [DecidableEq ι] |
| {κ : ι → Type*} (t : (j : ι) → Finset (κ j)) |
| (w : (j : ι) → κ j → ℝ) |
| (hw : ∀ j x, x ∈ t j → 0 ≤ w j x) |
| (P : ((j : ι) → κ j) → Prop) [DecidablePred P] : |
| (∑ x ∈ (Fintype.piFinset t).filter P, ∏ j, w j (x j)) ≤ |
| ∏ j, ∑ x ∈ t j, w j x := by |
| classical |
| rw [Finset.prod_univ_sum] |
| apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) |
| intro x hx _ |
| apply Finset.prod_nonneg |
| intro j _ |
| exact hw j (x j) ((Fintype.mem_piFinset.mp hx) j) |
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| / |
| def degreeWeight (q lam : ℝ) (e : ℕ) : ℝ := |
| ∑ i ∈ Finset.range (e + 1), q ^ |(i : ℝ) - lam * e| |
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| / |
| product of the individual degree weights. -/ |
| theorem bidegree_weight_sum_le {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (e : ι → ℕ) (k : ℕ) (q lam : ℝ) (hq : 0 ≤ q) : |
| (∑ i ∈ (Fintype.piFinset (fun j => Finset.range (e j + 1))).filter |
| (fun i => ∑ j, i j = k), |
| ∏ j, q ^ |(i j : ℝ) - lam * e j|) ≤ |
| ∏ j, degreeWeight q lam (e j) := by |
| classical |
| exact constrained_sum_prod_le_prod_sum |
| (fun j => Finset.range (e j + 1)) |
| (fun j i => q ^ |(i : ℝ) - lam * e j|) |
| (fun _ _ _ => Real.rpow_nonneg hq _) _ |
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| end Problem335 |
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| end |
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| end OAI |
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