import Erdos1135.Tao.Fourier.Section7SourceFThreeQFinite import Erdos1135.Tao.Fourier.Section7SourceLaw import Erdos1135.Tao.Fourier.Section7SourcePairing /-! # Section 7 Fixed-Sum Pair Fibers This leaf groups two independent positive Geom(2) valuations by their sum. The character numerator is left unnormalized, so the zero-mass fibers at source values `0` and `1` require no conditional division. -/ open scoped BigOperators namespace Erdos1135 namespace Tao /-- Two independent positive Geom(2) valuations before summing the pair. -/ noncomputable def taoSection7Geom2PairPMF : PMF (ℕ+ × ℕ+) := geom2PNat.bind fun a₁ => geom2PNat.map fun a₂ => (a₁, a₂) /-- The pair PMF pushes forward exactly to the raw Pascal source law. -/ theorem taoSection7Geom2PairPMF_map_sum_eq_pascalSourcePMF : taoSection7Geom2PairPMF.map (fun a => (a.1 : ℕ) + (a.2 : ℕ)) = taoSection7PascalSourcePMF := by unfold taoSection7Geom2PairPMF taoSection7PascalSourcePMF rw [PMF.map_bind] apply congrArg (fun f : ℕ+ → PMF ℕ => geom2PNat.bind f) funext a₁ rw [PMF.map_comp] change geom2PNat.map (fun a₂ : ℕ+ => (a₁ : ℕ) + (a₂ : ℕ)) = (geom2PNat.map fun a₂ : ℕ+ => (a₂ : ℕ)).map (fun b => (a₁ : ℕ) + b) rw [PMF.map_comp] rfl /-- Point mass of an ordered independent Geom(2) pair. -/ theorem taoSection7Geom2PairPMF_apply_toReal (a₁ a₂ : ℕ+) : (taoSection7Geom2PairPMF (a₁, a₂)).toReal = geom2PNatPairMass a₁ a₂ := by classical rw [taoSection7Geom2PairPMF, PMF.bind_apply] rw [tsum_eq_single a₁] · rw [PMF.map_apply] rw [tsum_eq_single a₂] · simp [geom2PNatPairMass, ENNReal.toReal_mul] · intro b hb have hpair : ¬(a₁, a₂) = (a₁, b) := by intro h exact hb (Prod.ext_iff.mp h).2.symm simp [hpair] · intro a ha have hmap_zero : (geom2PNat.map fun b => (a, b)) (a₁, a₂) = 0 := by rw [PMF.map_apply, ENNReal.tsum_eq_zero] intro b have hpair : ¬(a₁, a₂) = (a, b) := by intro h exact ha (Prod.ext_iff.mp h).1.symm simp [hpair] simp [hmap_zero] private def taoSection7FixedSumPairToIndex (b : ℕ) : {p : ℕ+ × ℕ+ // b = (p.1 : ℕ) + (p.2 : ℕ)} → {a : ℕ // a ∈ Finset.Icc 1 (b - 1)} := fun p => ⟨(p.1.1 : ℕ), by rw [Finset.mem_Icc] constructor · exact p.1.1.2 · have hp₂ : 0 < (p.1.2 : ℕ) := p.1.2.2 omega⟩ private theorem taoSection7FixedSumPairToIndex_injective (b : ℕ) : Function.Injective (taoSection7FixedSumPairToIndex b) := by rintro ⟨⟨a₁, a₂⟩, ha⟩ ⟨⟨c₁, c₂⟩, hc⟩ h have h₁ : (a₁ : ℕ) = (c₁ : ℕ) := congrArg (fun a : {a : ℕ // a ∈ Finset.Icc 1 (b - 1)} => (a : ℕ)) h have ha' : b = (a₁ : ℕ) + (a₂ : ℕ) := ha have hc' : b = (c₁ : ℕ) + (c₂ : ℕ) := hc have h₂ : (a₂ : ℕ) = (c₂ : ℕ) := by omega apply Subtype.ext apply Prod.ext · exact Subtype.ext h₁ · exact Subtype.ext h₂ private theorem taoSection7FixedSumPairToIndex_surjective (b : ℕ) : Function.Surjective (taoSection7FixedSumPairToIndex b) := by rintro ⟨a, ha⟩ rcases Finset.mem_Icc.mp ha with ⟨ha₁, hab⟩ have hab' : a ≤ b := le_trans hab (Nat.sub_le b 1) have hba : 0 < b - a := by omega refine ⟨⟨(⟨a, ha₁⟩, ⟨b - a, hba⟩), ?_⟩, ?_⟩ · exact (Nat.add_sub_of_le hab').symm · apply Subtype.ext rfl private noncomputable def taoSection7FixedSumPairEquiv (b : ℕ) : {p : ℕ+ × ℕ+ // b = (p.1 : ℕ) + (p.2 : ℕ)} ≃ {a : ℕ // a ∈ Finset.Icc 1 (b - 1)} := Equiv.ofBijective (taoSection7FixedSumPairToIndex b) ⟨taoSection7FixedSumPairToIndex_injective b, taoSection7FixedSumPairToIndex_surjective b⟩ private theorem taoSection7FixedSumPair_mass (b : ℕ) (p : {p : ℕ+ × ℕ+ // b = (p.1 : ℕ) + (p.2 : ℕ)}) : (taoSection7Geom2PairPMF p.1).toReal = (1 / 2 : ℝ) ^ (p.1.1 : ℕ) * (1 / 2 : ℝ) ^ (b - (p.1.1 : ℕ)) := by rw [taoSection7Geom2PairPMF_apply_toReal] unfold geom2PNatPairMass rw [geom2PNat_apply_toReal, geom2PNat_apply_toReal] have hsub : b - (p.1.1 : ℕ) = (p.1.2 : ℕ) := by omega rw [hsub] /-- The raw Pascal source mass is the finite fixed-sum pair mass, including the empty fibers at source values zero and one. -/ theorem taoSection7PascalSourcePMF_apply_toReal (b : ℕ) : (taoSection7PascalSourcePMF b).toReal = pascalGeom2PairMass b := by classical rw [← taoSection7Geom2PairPMF_map_sum_eq_pascalSourcePMF, pmf_map_apply_toReal_tsum] calc (∑' p : ℕ+ × ℕ+, if b = (p.1 : ℕ) + (p.2 : ℕ) then (taoSection7Geom2PairPMF p).toReal else 0) = ∑' p : {p : ℕ+ × ℕ+ // b = (p.1 : ℕ) + (p.2 : ℕ)}, (taoSection7Geom2PairPMF p.1).toReal := by simpa only [Set.indicator, Set.mem_setOf_eq] using (tsum_subtype {p : ℕ+ × ℕ+ | b = (p.1 : ℕ) + (p.2 : ℕ)} (fun p => (taoSection7Geom2PairPMF p).toReal)).symm _ = ∑' a : {a : ℕ // a ∈ Finset.Icc 1 (b - 1)}, (1 / 2 : ℝ) ^ a.1 * (1 / 2 : ℝ) ^ (b - a.1) := by rw [← (taoSection7FixedSumPairEquiv b).tsum_eq] exact tsum_congr (taoSection7FixedSumPair_mass b) _ = ∑ a ∈ Finset.Icc 1 (b - 1), (1 / 2 : ℝ) ^ a * (1 / 2 : ℝ) ^ (b - a) := by rw [tsum_fintype, ← Finset.attach_eq_univ] exact Finset.sum_attach (Finset.Icc 1 (b - 1)) (fun a => (1 / 2 : ℝ) ^ a * (1 / 2 : ℝ) ^ (b - a)) _ = pascalGeom2PairMass b := by rfl /-- The exact-length raw Pascal list law has the expected product point mass. -/ theorem taoSection7PascalSourceListPMF_apply_length_toReal_eq_pairMassProduct (bs : List ℕ) : (taoSection7PascalSourceListPMF bs.length bs).toReal = (bs.map pascalGeom2PairMass).prod := by induction bs with | nil => simp [taoSection7PascalSourceListPMF] | cons b bs ih => rw [List.length_cons, taoSection7PascalSourceListPMF_succ_apply_cons, ENNReal.toReal_mul, taoSection7PascalSourcePMF_apply_toReal, ih] simp /-- Raw character phase contributed by the second valuation in one pair. -/ noncomputable def taoSection7PairPhase (n : ℕ) (xi x : ZMod (3 ^ n)) (a₂ : ℕ+) : ℂ := taoForwardDFTKernel ((((2 : ZMod (3 ^ n)) ^ (a₂ : ℕ) + 3) * x)) xi @[simp] theorem taoSection7PairPhase_one (n : ℕ) (xi x : ZMod (3 ^ n)) : taoSection7PairPhase n xi x (1 : ℕ+) = taoForwardDFTKernel ((5 : ZMod (3 ^ n)) * x) xi := by unfold taoSection7PairPhase congr 2 norm_num @[simp] theorem taoSection7PairPhase_two (n : ℕ) (xi x : ZMod (3 ^ n)) : taoSection7PairPhase n xi x (2 : ℕ+) = taoForwardDFTKernel ((7 : ZMod (3 ^ n)) * x) xi := by simp [taoSection7PairPhase, pow_two] congr 2 ring /-- Unnormalized character numerator over the ordered positive pairs with sum `b`. -/ noncomputable def taoSection7PairFiberCharacterMass (n : ℕ) (xi x : ZMod (3 ^ n)) (b : ℕ) : ℂ := ∑' a : ℕ+ × ℕ+, if (a.1 : ℕ) + (a.2 : ℕ) = b then ((taoSection7Geom2PairPMF a).toReal : ℂ) * taoSection7PairPhase n xi x a.2 else 0 private theorem pnat_pair_eq_one_two_or_two_one_of_sum_eq_three (a₁ a₂ : ℕ+) (h : (a₁ : ℕ) + (a₂ : ℕ) = 3) : (a₁ = 1 ∧ a₂ = 2) ∨ (a₁ = 2 ∧ a₂ = 1) := by have ha₁pos : 0 < (a₁ : ℕ) := a₁.2 have ha₂pos : 0 < (a₂ : ℕ) := a₂.2 have ha₁ : (a₁ : ℕ) = 1 ∨ (a₁ : ℕ) = 2 := by omega rcases ha₁ with ha₁ | ha₁ · left constructor · apply Subtype.ext change (a₁ : ℕ) = 1 exact ha₁ · apply Subtype.ext change (a₂ : ℕ) = 2 omega · right constructor · apply Subtype.ext change (a₁ : ℕ) = 2 exact ha₁ · apply Subtype.ext change (a₂ : ℕ) = 1 omega private theorem taoSection7PairFiberCharacterMass_three_explicit (n : ℕ) (xi x : ZMod (3 ^ n)) : taoSection7PairFiberCharacterMass n xi x 3 = ((1 / 8 : ℝ) : ℂ) * taoForwardDFTKernel ((7 : ZMod (3 ^ n)) * x) xi + ((1 / 8 : ℝ) : ℂ) * taoForwardDFTKernel ((5 : ZMod (3 ^ n)) * x) xi := by classical unfold taoSection7PairFiberCharacterMass rw [tsum_eq_sum (s := ({((1 : ℕ+), (2 : ℕ+)), ((2 : ℕ+), (1 : ℕ+))} : Finset (ℕ+ × ℕ+)))] · simp [taoSection7Geom2PairPMF_apply_toReal, geom2PNatPairMass_one_two, geom2PNatPairMass_two_one, add_comm] · intro a ha by_cases hsum : (a.1 : ℕ) + (a.2 : ℕ) = 3 · rcases pnat_pair_eq_one_two_or_two_one_of_sum_eq_three a.1 a.2 hsum with h | h · exfalso apply ha simp only [Finset.mem_insert, Finset.mem_singleton] exact Or.inl (Prod.ext h.1 h.2) · exfalso apply ha simp only [Finset.mem_insert, Finset.mem_singleton] exact Or.inr (Prod.ext h.1 h.2) · simp [hsum] /-- The raw Pascal source mass at `b = 3` is `1/4`. -/ theorem taoSection7PascalSourcePMF_three_toReal : (taoSection7PascalSourcePMF 3).toReal = (1 / 4 : ℝ) := by rw [← taoSection7Geom2PairPMF_map_sum_eq_pascalSourcePMF] rw [pmf_map_apply_toReal_tsum] classical rw [tsum_eq_sum (s := ({((1 : ℕ+), (2 : ℕ+)), ((2 : ℕ+), (1 : ℕ+))} : Finset (ℕ+ × ℕ+)))] · simp [taoSection7Geom2PairPMF_apply_toReal, geom2PNatPairMass_one_two, geom2PNatPairMass_two_one] norm_num · intro a ha by_cases hsum : (a.1 : ℕ) + (a.2 : ℕ) = 3 · rcases pnat_pair_eq_one_two_or_two_one_of_sum_eq_three a.1 a.2 hsum with h | h · exfalso apply ha simp only [Finset.mem_insert, Finset.mem_singleton] exact Or.inl (Prod.ext h.1 h.2) · exfalso apply ha simp only [Finset.mem_insert, Finset.mem_singleton] exact Or.inr (Prod.ext h.1 h.2) · have hsum' : ¬3 = (a.1 : ℕ) + (a.2 : ℕ) := by omega simp [hsum'] /-- At the cancellation atom `b = 3`, the unnormalized numerator is exactly the raw source mass times Tao's `FThree`. The source orientation is `(1,2) ↦ 7x` and `(2,1) ↦ 5x`; equal pair masses identify it with the existing symmetrically weighted `FThree`. -/ theorem taoSection7PairFiberCharacterMass_three_eq (n : ℕ) (xi x : ZMod (3 ^ n)) : taoSection7PairFiberCharacterMass n xi x 3 = ((taoSection7PascalSourcePMF 3).toReal : ℂ) * taoSection7FThree n xi x := by rw [taoSection7PairFiberCharacterMass_three_explicit, taoSection7PascalSourcePMF_three_toReal] unfold taoSection7FThree rw [geom2PNatPairMass_one_two_div_pascal_three, geom2PNatPairMass_two_one_div_pascal_three] simp only [Complex.ofReal_div, Complex.ofReal_one, Complex.ofReal_ofNat] ring private theorem taoSection7PairFiberCharacterMass_summable (n : ℕ) (xi x : ZMod (3 ^ n)) (b : ℕ) : Summable fun a : ℕ+ × ℕ+ => if (a.1 : ℕ) + (a.2 : ℕ) = b then ((taoSection7Geom2PairPMF a).toReal : ℂ) * taoSection7PairPhase n xi x a.2 else 0 := by refine Summable.of_norm_bounded (pmf_summable_toReal taoSection7Geom2PairPMF) ?_ intro a by_cases hsum : (a.1 : ℕ) + (a.2 : ℕ) = b · have hphase : ‖taoSection7PairPhase n xi x a.2‖ = 1 := by unfold taoSection7PairPhase exact norm_taoForwardDFTKernel_eq_one _ _ simp only [hsum, if_true, norm_mul, Complex.norm_real, hphase, mul_one] simp [abs_of_nonneg ENNReal.toReal_nonneg] · simp [hsum, ENNReal.toReal_nonneg] /-- Every fixed-sum character numerator is bounded by the mass of its Pascal fiber. -/ theorem norm_taoSection7PairFiberCharacterMass_le_sourceMass (n : ℕ) (xi x : ZMod (3 ^ n)) (b : ℕ) : ‖taoSection7PairFiberCharacterMass n xi x b‖ ≤ (taoSection7PascalSourcePMF b).toReal := by rw [taoSection7PairFiberCharacterMass] calc ‖∑' a : ℕ+ × ℕ+, if (a.1 : ℕ) + (a.2 : ℕ) = b then ((taoSection7Geom2PairPMF a).toReal : ℂ) * taoSection7PairPhase n xi x a.2 else 0‖ ≤ ∑' a : ℕ+ × ℕ+, ‖if (a.1 : ℕ) + (a.2 : ℕ) = b then ((taoSection7Geom2PairPMF a).toReal : ℂ) * taoSection7PairPhase n xi x a.2 else 0‖ := norm_tsum_le_tsum_norm (taoSection7PairFiberCharacterMass_summable n xi x b).norm _ = ∑' a : ℕ+ × ℕ+, if b = (a.1 : ℕ) + (a.2 : ℕ) then (taoSection7Geom2PairPMF a).toReal else 0 := by apply tsum_congr intro a by_cases hsum : (a.1 : ℕ) + (a.2 : ℕ) = b · have hphase : ‖taoSection7PairPhase n xi x a.2‖ = 1 := by unfold taoSection7PairPhase exact norm_taoForwardDFTKernel_eq_one _ _ simp [hsum, hphase] · have hsum' : ¬b = (a.1 : ℕ) + (a.2 : ℕ) := by omega simp [hsum, hsum'] _ = (taoSection7PascalSourcePMF b).toReal := by rw [← taoSection7Geom2PairPMF_map_sum_eq_pascalSourcePMF] exact (pmf_map_apply_toReal_tsum taoSection7Geom2PairPMF (fun a => (a.1 : ℕ) + (a.2 : ℕ)) b).symm /-- Unnormalized pair-fiber bound in the source `(j,s,b)` coordinates. -/ theorem norm_taoSection7PairFiberCharacterMass_le_sourceFactor (n : ℕ) (xi : ZMod (3 ^ n)) (j : ℕ+) (s b : ℕ) : ‖taoSection7PairFiberCharacterMass n xi (taoSection7PairX n j (Int.ofNat (s + b))) b‖ ≤ (taoSection7PascalSourcePMF b).toReal * taoSection7SourceFThreeFactor n xi (j : ℕ) s b := by by_cases hb : b = 3 · subst b rw [taoSection7PairFiberCharacterMass_three_eq] have hcond : (3 : ℕ) = 3 ∧ 0 < (j : ℕ) := ⟨rfl, j.2⟩ rw [taoSection7SourceFThreeFactor, dif_pos hcond] rw [norm_mul, Complex.norm_real] have hj : (⟨(j : ℕ), hcond.2⟩ : ℕ+) = j := Subtype.ext rfl rw [hj] simp · rw [taoSection7SourceFThreeFactor, dif_neg (by tauto)] simpa using norm_taoSection7PairFiberCharacterMass_le_sourceMass n xi (taoSection7PairX n j (Int.ofNat (s + b))) b end Tao end Erdos1135