| import OAI.NumberTheory.Ostmann.Construction.RegularDiagonalCancellation |
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| namespace OAI |
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| /-! # Removing the regular-prime block from the signed arithmetic average -/ |
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| namespace Ostmann |
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| open scoped BigOperators |
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| / |
| noncomputable def crtSplitExternalEquiv (c r : ℕ) (hc : c.Coprime r) : |
| (ZMod (c * r) × (ZMod (c * r))ˣ) ≃ |
| (ZMod c × (ZMod c)ˣ) × (ZMod r × (ZMod r)ˣ) := |
| ((ZMod.chineseRemainder hc).toEquiv.prodCongr (crtSplitUnitEquiv c r hc)).trans |
| (Equiv.prodProdProdComm _ _ _ _) |
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| / |
| tests, including the retained Page correction. No absolute value is taken. -/ |
| theorem crt_external_regular_with_remainder {I : Type*} [Fintype I] [DecidableEq I] |
| (p : I → ℕ) [∀ i, Fact (p i).Prime] [∀ i, NeZero (p i)] [NeZero (∏ i, p i)] |
| (hc : Pairwise (fun i j => (p i).Coprime (p j))) |
| (r : ℕ) [NeZero r] [NeZero ((∏ i, p i) * r)] |
| (hcop : (∏ i, p i).Coprime r) |
| (active : I → Bool) (a : ∀ i, (ZMod (p i))ˣ) (g : ∀ i, ZMod (p i) → ℂ) |
| (hg : ∀ i, g i 0 = 0) (henergy : ∀ i, (∑ x : ZMod (p i), ‖g i x‖ ^ 2) = p i) |
| (F : ZMod r × (ZMod r)ˣ → ℂ) : |
| (Fintype.card (ZMod ((∏ i, p i) * r) × (ZMod ((∏ i, p i) * r))ˣ) : ℂ)⁻¹ * |
| (∑ z, |
| ((∏ i, externalRegularLocal (active i) (a i) (g i) |
| (crtExternalPairEquiv p hc (crtSplitExternalEquiv (∏ i, p i) r hcop z).1 i) : ℝ) : ℂ) * |
| F (crtSplitExternalEquiv (∏ i, p i) r hcop z).2) = |
| ((∏ i, if active i then (1 : ℝ) else 1 - (p i : ℝ)⁻¹ : ℝ) : ℂ) * |
| ((Fintype.card (ZMod r × (ZMod r)ˣ) : ℂ)⁻¹ * ∑ z, F z) := by |
| have hreg := crt_external_regular_cancellation p hc active a g hg henergy |
| have hregC : (Fintype.card (ZMod (∏ i, p i) × (ZMod (∏ i, p i))ˣ) : ℂ)⁻¹ * |
| (∑ z, ((∏ i, externalRegularLocal (active i) (a i) (g i) |
| (crtExternalPairEquiv p hc z i) : ℝ) : ℂ)) = |
| ((∏ i, if active i then (1 : ℝ) else 1 - (p i : ℝ)⁻¹ : ℝ) : ℂ) := by |
| simpa only [Complex.ofReal_mul, Complex.ofReal_inv, Complex.ofReal_natCast, |
| Complex.ofReal_sum] using congrArg (fun t : ℝ => (t : ℂ)) hreg |
| rw [uniform_equiv_product_average (crtSplitExternalEquiv (∏ i, p i) r hcop) |
| (fun z => ((∏ i, externalRegularLocal (active i) (a i) (g i) |
| (crtExternalPairEquiv p hc z i) : ℝ) : ℂ)) F, hregC] |
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| end Ostmann |
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| end OAI |
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