import Erdos1135.Tao.Fourier.Section7Character import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds import Mathlib.Analysis.Complex.Trigonometric import Mathlib.Data.ZMod.ValMinAbs import Mathlib.Tactic /-! # Section 7 Local Cancellation This module records the local numerical cancellation used at the white-point stage of Tao's Section 7, Lemma 7.2. It does not define white points, prove Lemma 7.2, or prove any renewal estimate. -/ open scoped ZMod namespace Erdos1135 namespace Tao /-- Tao-style signed theta representative of a residue modulo `N`. -/ noncomputable def taoSignedTheta {N : ℕ} [NeZero N] (z : ZMod N) : ℝ := (z.valMinAbs : ℝ) / (N : ℝ) /-- The signed representative casts back to its residue. -/ theorem taoSignedTheta_intCast_valMinAbs {N : ℕ} [NeZero N] (z : ZMod N) : ((z.valMinAbs : ℤ) : ZMod N) = z := by exact ZMod.coe_valMinAbs z /-- Source-faithful black predicate from Tao (7.9): `|theta| <= epsilon`. -/ def taoSection7Black {N : ℕ} [NeZero N] (epsilon : ℝ) (z : ZMod N) : Prop := |taoSignedTheta z| ≤ epsilon /-- Source-faithful white predicate: the complement of black, `epsilon < |theta|`. -/ def taoSection7White {N : ℕ} [NeZero N] (epsilon : ℝ) (z : ZMod N) : Prop := epsilon < |taoSignedTheta z| theorem taoSection7White_iff_not_black {N : ℕ} [NeZero N] (epsilon : ℝ) (z : ZMod N) : taoSection7White epsilon z ↔ ¬ taoSection7Black epsilon z := by simp [taoSection7White, taoSection7Black, not_le] /-- The signed representative lies in the half interval used by Tao's `{}`. -/ theorem abs_taoSignedTheta_le_half {N : ℕ} [NeZero N] (z : ZMod N) : |taoSignedTheta z| ≤ 1 / 2 := by have hIoc := ZMod.valMinAbs_mem_Ioc z have hNpos_real : (0 : ℝ) < (N : ℝ) := by exact_mod_cast NeZero.pos N have hupper : ((z.valMinAbs : ℝ) * 2) ≤ (N : ℝ) := by exact_mod_cast hIoc.2 have hlower : (-(N : ℝ)) < ((z.valMinAbs : ℝ) * 2) := by exact_mod_cast hIoc.1 rw [taoSignedTheta, abs_div, abs_of_pos hNpos_real] rw [div_le_iff₀ hNpos_real] rw [abs_le] constructor <;> linarith /-- Negative-sign additive character on `ZMod N`, with a real-ratio exponent. -/ theorem stdAddChar_neg_intCast_realRatio {N : ℕ} [NeZero N] (m : ℤ) : ZMod.stdAddChar (-(m : ZMod N)) = Complex.exp (-(2 * Real.pi * ((m : ℝ) / (N : ℝ))) * Complex.I) := by have h := ZMod.stdAddChar_coe (N := N) (-m) rw [show (-(m : ZMod N)) = ((-m : ℤ) : ZMod N) by norm_num] rw [h] congr 1 norm_num field_simp [NeZero.ne N] /-- Natural-number specialization of the negative-sign convention bridge. -/ theorem stdAddChar_neg_natCast_realRatio {N : ℕ} [NeZero N] (m : ℕ) : ZMod.stdAddChar (-(m : ZMod N)) = Complex.exp (-(2 * Real.pi * ((m : ℝ) / (N : ℝ))) * Complex.I) := by simpa using stdAddChar_neg_intCast_realRatio (N := N) (m : ℤ) /-- Direct expression of the negative additive character through `taoSignedTheta`. -/ theorem stdAddChar_neg_residue_eq_exp_taoSignedTheta {N : ℕ} [NeZero N] (z : ZMod N) : ZMod.stdAddChar (-z) = Complex.exp (-(2 * Real.pi * taoSignedTheta z) * Complex.I) := by rw [← taoSignedTheta_intCast_valMinAbs z] rw [stdAddChar_neg_intCast_realRatio (N := N) z.valMinAbs] simp [taoSignedTheta] /-- A real-variable front end for Tao Lemma 7.2's final white-point estimate. The source theta is a signed fractional-part representative in `(-1/2, 1/2]`. This lemma intentionally assumes only the weaker bound `|theta| <= 1`. -/ theorem tao_cos_pi_le_exp_neg_cube_of_white {epsilon theta : ℝ} (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) (hwhite : epsilon ≤ |theta|) (htheta : |theta| ≤ 1) : Real.cos (Real.pi * theta) ≤ Real.exp (-(epsilon ^ 3)) := by have hpi_nonneg : 0 ≤ Real.pi := Real.pi_pos.le have harg : |Real.pi * theta| ≤ Real.pi := by rw [abs_mul, abs_of_nonneg hpi_nonneg] exact mul_le_of_le_one_right hpi_nonneg htheta have hcos := Real.cos_le_one_sub_mul_cos_sq harg have hpi_ne : Real.pi ≠ 0 := Real.pi_ne_zero have hrewrite : 1 - 2 / Real.pi ^ 2 * (Real.pi * theta) ^ 2 = 1 - 2 * theta ^ 2 := by field_simp [hpi_ne] have hsq_abs : |theta| ^ 2 = theta ^ 2 := by rw [sq_abs] have hsq : epsilon ^ 2 ≤ theta ^ 2 := by have := pow_le_pow_left₀ hepsilon0 hwhite 2 rwa [hsq_abs] at this have hepsilon_le_two : epsilon ≤ 2 := by linarith have hcube_le : epsilon ^ 3 ≤ 2 * epsilon ^ 2 := by nlinarith [mul_le_mul_of_nonneg_right hepsilon_le_two (sq_nonneg epsilon)] have hpoly : 1 - 2 * theta ^ 2 ≤ 1 - epsilon ^ 3 := by nlinarith have hexp : 1 - epsilon ^ 3 ≤ Real.exp (-(epsilon ^ 3)) := by have := Real.add_one_le_exp (-(epsilon ^ 3)) linarith calc Real.cos (Real.pi * theta) ≤ 1 - 2 / Real.pi ^ 2 * (Real.pi * theta) ^ 2 := hcos _ = 1 - 2 * theta ^ 2 := hrewrite _ ≤ 1 - epsilon ^ 3 := hpoly _ ≤ Real.exp (-(epsilon ^ 3)) := hexp /-- Complex half-sum identity in the representative range used by Tao's signed fractional part. -/ theorem norm_half_one_add_exp_neg_two_pi_i {theta : ℝ} (htheta : |theta| ≤ 1 / 2) : ‖(1 / 2 : ℂ) * (1 + Complex.exp (-(2 * Real.pi * theta) * Complex.I))‖ = Real.cos (Real.pi * theta) := by let x : ℝ := Real.pi * theta have hhalf : |x| ≤ Real.pi / 2 := by rw [show x = Real.pi * theta by rfl, abs_mul, abs_of_nonneg Real.pi_pos.le] nlinarith [mul_le_mul_of_nonneg_left htheta Real.pi_pos.le] have hcos_nonneg : 0 ≤ Real.cos x := Real.cos_nonneg_of_mem_Icc (abs_le.mp hhalf) have hnorm : ‖(1 / 2 : ℂ) * (1 + Complex.exp (-(2 * x) * Complex.I))‖ = |Real.cos x| := by have hsum : Complex.exp ((x : ℂ) * Complex.I) + Complex.exp (-(x : ℂ) * Complex.I) = 2 * (Real.cos x : ℂ) := by simpa [Complex.ofReal_cos] using (Complex.two_cos (x : ℂ)).symm have hdecomp : 1 + Complex.exp (-(2 * x) * Complex.I) = Complex.exp (-(x : ℂ) * Complex.I) * (Complex.exp ((x : ℂ) * Complex.I) + Complex.exp (-(x : ℂ) * Complex.I)) := by rw [eq_comm] calc Complex.exp (-(x : ℂ) * Complex.I) * (Complex.exp ((x : ℂ) * Complex.I) + Complex.exp (-(x : ℂ) * Complex.I)) = Complex.exp (-(x : ℂ) * Complex.I) * Complex.exp ((x : ℂ) * Complex.I) + Complex.exp (-(x : ℂ) * Complex.I) * Complex.exp (-(x : ℂ) * Complex.I) := by ring _ = Complex.exp (-(x : ℂ) * Complex.I + (x : ℂ) * Complex.I) + Complex.exp (-(x : ℂ) * Complex.I + (-(x : ℂ) * Complex.I)) := by rw [Complex.exp_add, Complex.exp_add] _ = 1 + Complex.exp (-(2 * x) * Complex.I) := by congr 1 · ring_nf simp · congr 1 norm_num ring calc ‖(1 / 2 : ℂ) * (1 + Complex.exp (-(2 * x) * Complex.I))‖ = ‖(1 / 2 : ℂ) * (Complex.exp (-(x : ℂ) * Complex.I) * (Complex.exp ((x : ℂ) * Complex.I) + Complex.exp (-(x : ℂ) * Complex.I)))‖ := by rw [hdecomp] _ = ‖Complex.exp (-(x : ℂ) * Complex.I) * ((1 / 2 : ℂ) * (Complex.exp ((x : ℂ) * Complex.I) + Complex.exp (-(x : ℂ) * Complex.I)))‖ := by congr 1 ring _ = ‖Complex.exp (-(x : ℂ) * Complex.I) * ((1 / 2 : ℂ) * (2 * (Real.cos x : ℂ)))‖ := by rw [hsum] _ = ‖Complex.exp (-(x : ℂ) * Complex.I) * (Real.cos x : ℂ)‖ := by congr 1 ring _ = |Real.cos x| := by rw [norm_mul] rw [show -(x : ℂ) * Complex.I = ((-x : ℝ) : ℂ) * Complex.I by norm_num] rw [Complex.norm_exp_ofReal_mul_I, one_mul, Complex.norm_real, Real.norm_eq_abs] simpa [x, abs_of_nonneg hcos_nonneg, mul_assoc] using hnorm /-- Combined post-theta estimate for the local `b = 3` cancellation step. This does not connect the project `ZMod.stdAddChar` convention to the source theta; the direct integer-representative theorem below supplies that local bridge. -/ theorem norm_half_one_add_exp_neg_two_pi_i_le_exp_neg_cube_of_white {epsilon theta : ℝ} (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) (hwhite : epsilon ≤ |theta|) (htheta : |theta| ≤ 1 / 2) : ‖(1 / 2 : ℂ) * (1 + Complex.exp (-(2 * Real.pi * theta) * Complex.I))‖ ≤ Real.exp (-(epsilon ^ 3)) := by rw [norm_half_one_add_exp_neg_two_pi_i htheta] exact tao_cos_pi_le_exp_neg_cube_of_white hepsilon0 hepsilon1 hwhite (by nlinarith) /-- Direct local estimate for a `ZMod.stdAddChar` term once an integer representative `m` and Tao's signed fractional representative `theta = m / N` have been chosen. -/ theorem norm_half_one_add_stdAddChar_neg_intCast_le_exp_neg_cube_of_white {N : ℕ} [NeZero N] {m : ℤ} {epsilon theta : ℝ} (htheta_eq : theta = (m : ℝ) / (N : ℝ)) (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) (hwhite : epsilon ≤ |theta|) (htheta_half : |theta| ≤ 1 / 2) : ‖(1 / 2 : ℂ) * (1 + ZMod.stdAddChar (-(m : ZMod N)))‖ ≤ Real.exp (-(epsilon ^ 3)) := by rw [stdAddChar_neg_intCast_realRatio (N := N) m] have hratio : ((m : ℂ) / (N : ℂ)) = (theta : ℂ) := by rw [htheta_eq] norm_cast rw [show Complex.exp (-(2 * Real.pi * ((m : ℝ) / (N : ℝ))) * Complex.I) = Complex.exp (-(2 * Real.pi * theta) * Complex.I) by congr 1 change -(2 * ↑Real.pi * ((m : ℂ) / (N : ℂ))) * Complex.I = -(2 * ↑Real.pi * (theta : ℂ)) * Complex.I rw [hratio]] rw [norm_half_one_add_exp_neg_two_pi_i htheta_half] exact tao_cos_pi_le_exp_neg_cube_of_white hepsilon0 hepsilon1 hwhite (by nlinarith) /-- Canonical predicate-facing local white-point cancellation bridge. The strict source white condition is used only through its weak consequence `epsilon <= |theta|`. -/ theorem norm_half_one_add_stdAddChar_neg_of_taoSection7White {N : ℕ} [NeZero N] (z : ZMod N) {epsilon : ℝ} (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) (hwhite : taoSection7White epsilon z) : ‖(1 / 2 : ℂ) * (1 + ZMod.stdAddChar (-z))‖ ≤ Real.exp (-(epsilon ^ 3)) := by rw [stdAddChar_neg_residue_eq_exp_taoSignedTheta z] rw [norm_half_one_add_exp_neg_two_pi_i (abs_taoSignedTheta_le_half z)] exact tao_cos_pi_le_exp_neg_cube_of_white hepsilon0 hepsilon1 hwhite.le (by nlinarith [abs_taoSignedTheta_le_half z]) /-- Kernel-facing version of the white-point cancellation bound, matching the factor left after `norm_taoSection7PairAverage_eq`. -/ theorem norm_half_one_add_taoForwardDFTKernel_of_taoSection7White {N : ℕ} [NeZero N] (x xi : ZMod N) {epsilon : ℝ} (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) (hwhite : taoSection7White epsilon (((2 : ZMod N) * x) * xi)) : ‖(1 / 2 : ℂ) * (1 + taoForwardDFTKernel ((2 : ZMod N) * x) xi)‖ ≤ Real.exp (-(epsilon ^ 3)) := by simpa [taoForwardDFTKernel] using norm_half_one_add_stdAddChar_neg_of_taoSection7White (((2 : ZMod N) * x) * xi) hepsilon0 hepsilon1 hwhite /-- Direct local consumer for Tao's two-point pair average at `b = 3`. -/ theorem norm_taoSection7PairAverage_le_exp_neg_cube_of_taoSection7White (n : ℕ) (xi x : ZMod (3 ^ n)) {epsilon : ℝ} (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) (hwhite : taoSection7White epsilon (((2 : ZMod (3 ^ n)) * x) * xi)) : ‖taoSection7PairAverage n xi x‖ ≤ Real.exp (-(epsilon ^ 3)) := by rw [norm_taoSection7PairAverage_eq] exact norm_half_one_add_taoForwardDFTKernel_of_taoSection7White (N := 3 ^ n) x xi hepsilon0 hepsilon1 hwhite /-- The unit represented by `2` in `ZMod (3^n)`. -/ noncomputable def taoSection7TwoUnit (n : ℕ) : (ZMod (3 ^ n))ˣ := ZMod.unitOfCoprime 2 (Nat.Coprime.pow_right n (by decide : Nat.Coprime 2 3)) /-- Integer powers of `2` in `ZMod (3^n)`, via the unit group. -/ noncomputable def taoSection7TwoZPow (n : ℕ) (e : ℤ) : ZMod (3 ^ n) := ((taoSection7TwoUnit n) ^ e : (ZMod (3 ^ n))ˣ) theorem taoSection7TwoUnit_coe (n : ℕ) : (taoSection7TwoUnit n : ZMod (3 ^ n)) = 2 := by simp [taoSection7TwoUnit] theorem two_mul_taoSection7TwoZPow_neg (n : ℕ) (l : ℤ) : (2 : ZMod (3 ^ n)) * taoSection7TwoZPow n (-l) = taoSection7TwoZPow n (1 - l) := by unfold taoSection7TwoZPow rw [← taoSection7TwoUnit_coe n] rw [← Units.val_mul] congr 1 rw [show (1 - l : ℤ) = 1 + -l by ring] rw [zpow_add, zpow_one, zpow_neg] /-- The source `x = 3^(2j-2) * 2^(-l)` from the proof of Lemma 7.2. -/ noncomputable def taoSection7PairX (n : ℕ) (j : ℕ+) (l : ℤ) : ZMod (3 ^ n) := (3 : ZMod (3 ^ n)) ^ (2 * ((j : ℕ) - 1)) * taoSection7TwoZPow n (-l) /-- The residue whose signed representative is Tao's theta in equation (7.8): `xi * 3^(2j-2) * 2^(-l+1)` modulo `3^n`. -/ noncomputable def taoSection7ThetaResidue (n : ℕ) (xi : ZMod (3 ^ n)) (j : ℕ+) (l : ℤ) : ZMod (3 ^ n) := ((3 : ZMod (3 ^ n)) ^ (2 * ((j : ℕ) - 1)) * taoSection7TwoZPow n (1 - l)) * xi /-- Bridge from the pair-average kernel factor `chi(2x)` to Tao's point theta residue. -/ theorem taoSection7_pairX_thetaResidue (n : ℕ) (xi : ZMod (3 ^ n)) (j : ℕ+) (l : ℤ) : ((2 : ZMod (3 ^ n)) * taoSection7PairX n j l) * xi = taoSection7ThetaResidue n xi j l := by dsimp [taoSection7PairX, taoSection7ThetaResidue] calc ((2 : ZMod (3 ^ n)) * ((3 : ZMod (3 ^ n)) ^ (2 * ((j : ℕ) - 1)) * taoSection7TwoZPow n (-l))) * xi = (((3 : ZMod (3 ^ n)) ^ (2 * ((j : ℕ) - 1)) * ((2 : ZMod (3 ^ n)) * taoSection7TwoZPow n (-l))) * xi) := by ring _ = ((3 : ZMod (3 ^ n)) ^ (2 * ((j : ℕ) - 1)) * taoSection7TwoZPow n (1 - l)) * xi := by rw [two_mul_taoSection7TwoZPow_neg] /-- Point-level local Lemma 7.2 consumer: whiteness of Tao's theta residue at `(j,l)` bounds the pair average at `x = 3^(2j-2) * 2^(-l)`. -/ theorem norm_taoSection7PairAverage_point_le_exp_neg_cube_of_white (n : ℕ) (xi : ZMod (3 ^ n)) (j : ℕ+) (l : ℤ) {epsilon : ℝ} (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) (hwhite : taoSection7White epsilon (taoSection7ThetaResidue n xi j l)) : ‖taoSection7PairAverage n xi (taoSection7PairX n j l)‖ ≤ Real.exp (-(epsilon ^ 3)) := by rw [norm_taoSection7PairAverage_eq] refine norm_half_one_add_taoForwardDFTKernel_of_taoSection7White (N := 3 ^ n) (taoSection7PairX n j l) xi hepsilon0 hepsilon1 ?_ simpa [taoSection7_pairX_thetaResidue] using hwhite /-- Source-shaped conditional expectation `f(x,3)` from Tao equation (7.4). The two terms are the only ordered positive pairs with `a1 + a2 = 3`, weighted by their exact conditional masses from the local Pascal pair calculation. -/ noncomputable def taoSection7FThree (n : ℕ) (xi x : ZMod (3 ^ n)) : ℂ := (((geom2PNatPairMass (1 : ℕ+) (2 : ℕ+) / pascalGeom2PairMass 3 : ℝ) : ℂ) * taoForwardDFTKernel ((5 : ZMod (3 ^ n)) * x) xi) + (((geom2PNatPairMass (2 : ℕ+) (1 : ℕ+) / pascalGeom2PairMass 3 : ℝ) : ℂ) * taoForwardDFTKernel ((7 : ZMod (3 ^ n)) * x) xi) /-- Tao equation (7.4)'s conditional expectation at `b = 3` is the pair average. -/ theorem taoSection7FThree_eq_pairAverage (n : ℕ) (xi x : ZMod (3 ^ n)) : taoSection7FThree n xi x = taoSection7PairAverage n xi x := by unfold taoSection7FThree taoSection7PairAverage rw [geom2PNatPairMass_one_two_div_pascal_three] rw [geom2PNatPairMass_two_one_div_pascal_three] norm_num theorem norm_taoSection7FThree_eq_pairAverage (n : ℕ) (xi x : ZMod (3 ^ n)) : ‖taoSection7FThree n xi x‖ = ‖taoSection7PairAverage n xi x‖ := by rw [taoSection7FThree_eq_pairAverage] theorem norm_taoSection7FThree_le_of_pairAverage (n : ℕ) (xi x : ZMod (3 ^ n)) {B : ℝ} (hpair : ‖taoSection7PairAverage n xi x‖ ≤ B) : ‖taoSection7FThree n xi x‖ ≤ B := by rwa [norm_taoSection7FThree_eq_pairAverage] /-- Source-facing local Lemma 7.2 endpoint: if Tao's theta residue at `(j,l)` is white, then the source-shaped `f(3^(2j-2)2^(-l),3)` is bounded by `exp(-epsilon^3)`. -/ theorem norm_taoSection7FThree_point_le_exp_neg_cube_of_white (n : ℕ) (xi : ZMod (3 ^ n)) (j : ℕ+) (l : ℤ) {epsilon : ℝ} (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) (hwhite : taoSection7White epsilon (taoSection7ThetaResidue n xi j l)) : ‖taoSection7FThree n xi (taoSection7PairX n j l)‖ ≤ Real.exp (-(epsilon ^ 3)) := by rw [taoSection7FThree_eq_pairAverage] exact norm_taoSection7PairAverage_point_le_exp_neg_cube_of_white n xi j l hepsilon0 hepsilon1 hwhite /-- The same local estimate stated through the project DFT kernel at unit frequency, making the dependence on the R5 convention spine explicit. -/ theorem norm_half_one_add_taoForwardDFTKernel_intCast_one_le_exp_neg_cube_of_white {N : ℕ} [NeZero N] {m : ℤ} {epsilon theta : ℝ} (htheta_eq : theta = (m : ℝ) / (N : ℝ)) (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) (hwhite : epsilon ≤ |theta|) (htheta_half : |theta| ≤ 1 / 2) : ‖(1 / 2 : ℂ) * (1 + taoForwardDFTKernel (m : ZMod N) 1)‖ ≤ Real.exp (-(epsilon ^ 3)) := by simpa [taoForwardDFTKernel] using norm_half_one_add_stdAddChar_neg_intCast_le_exp_neg_cube_of_white (N := N) (m := m) (epsilon := epsilon) (theta := theta) htheta_eq hepsilon0 hepsilon1 hwhite htheta_half end Tao end Erdos1135