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a72760f | 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 | import OAI.NumberTheory.Ostmann.Construction.ArrangementProducts
import OAI.NumberTheory.Ostmann.Characters.TreeComparison
namespace OAI
/-! # Applying the quartet comparison to actual overlap components -/
namespace Ostmann
open scoped BigOperators Classical
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
/-- All component maps and their nonempty fibers are obtained from the
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
end Ostmann
end OAI
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