| import OAI.NumberTheory.Ostmann.Construction.ArrangementProducts |
| import OAI.NumberTheory.Ostmann.Characters.TreeComparison |
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|
| namespace OAI |
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| /-! # Applying the quartet comparison to actual overlap components -/ |
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| namespace Ostmann |
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| open scoped BigOperators Classical |
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| theorem arrangement_tree_component_count {n m : ℕ} |
| (e : Equiv.Perm (TreeLeafIndex (n + 2) × Fin m)) |
| (hgood : 4 * Fintype.card (arrangementGraph m e).ConnectedComponent ≤ |
| 3 * Fintype.card (TreeLeafIndex (n + 2))) : |
| Fintype.card (arrangementGraph m e).ConnectedComponent ≤ 3 * 2 ^ n := by |
| rw [card_treeLeafIndex, pow_add] at hgood |
| norm_num at hgood |
| omega |
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|
| / |
| actual bulk permutation; only the local Fourier bound is supplied. -/ |
| theorem arrangement_tree_comparison {p : ℕ} [Fact p.Prime] |
| (hp : 3 ≤ p) (g : ZMod p → ℂ) (hg : g 0 = 0) |
| (henergy : (∑ x : ZMod p, ‖g x‖ ^ 2) ≤ (p : ℝ)) |
| (ε : ℝ) (hε : 0 ≤ ε) (hflat : MixedFourierBound g ε) |
| (n m : ℕ) (hm : 0 < m) |
| (e : Equiv.Perm (TreeLeafIndex (n + 2) × Fin m)) |
| (hgood : 4 * Fintype.card (arrangementGraph m e).ConnectedComponent ≤ |
| 3 * Fintype.card (TreeLeafIndex (n + 2))) |
| (D₁ D₂ : (ZMod p)ˣ) {K₁ K₂ : (ZMod p)ˣ} |
| (T₁ : RationalTreeData (ZMod p)ˣ (n + 2) K₁) |
| (T₂ : RationalTreeData (ZMod p)ˣ (n + 2) K₂) |
| (XL₁ XR₁ XL₂ XR₂ : (ZMod p)ˣ) (c₁ c₂ : TreeLeafTuple Bool (n + 2)) |
| (cL cR : TreeLeafIndex n → Bool) |
| (hconj : ∀ q, quartetBlockEquiv Bool n c₁ q = ((cL q, !(cL q)), (cR q, !(cR q)))) : |
| ‖pairedPartitionCorrelation (arrangementLeft e) (arrangementRight e) |
| (indexedRationalAmplitude g D₁ T₁ XL₁ XR₁ c₁) |
| (indexedRationalAmplitude g D₂ T₂ XL₂ XR₂ c₂)‖ ^ 2 ≤ |
| (3 : ℝ) ^ (2 ^ (n + 2)) * ((2 : ℝ) ^ (2 ^ n - 1) * (quartetLocalMeanError p ε * |
| (8 * ((p : ℝ) / (Fintype.card (ZMod p)ˣ : ℝ)) ^ 4) ^ (2 ^ n - 1))) := by |
| choose pivot₁ hpivot₁ using arrangementLeft_surjective hm e |
| choose pivot₂ hpivot₂ using arrangementRight_surjective hm e |
| exact pairedRationalTree_comparison hp g hg henergy ε hε hflat n |
| (arrangementLeft e) (arrangementRight e) |
| (fun c => ⟨pivot₁ c, hpivot₁ c⟩) (fun c => ⟨pivot₂ c, hpivot₂ c⟩) |
| (arrangement_tree_component_count e hgood) |
| D₁ D₂ T₁ T₂ XL₁ XR₁ XL₂ XR₂ c₁ c₂ cL cR hconj |
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| end Ostmann |
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| end OAI |
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