import Erdos1135.Tao.Section6.ConductorIndex import Erdos1135.Tao.Section6.Corollary63 import Mathlib.Data.Nat.Factorization.Basic /-! # Section 6 Frequency Factorization This leaf factors a discarded frequency into its exact power of three and a primitive cofactor. It also records that unit scaling, including Tao's `2^{-l}` tail scaling, does not change membership in a power-of-three frequency subgroup. -/ namespace Erdos1135 namespace Tao noncomputable section /-- Frequencies divisible by `3^q` inside the ambient group `ZMod (3^T)`. -/ def zmodThreePowMultiple (T q : ℕ) (omega : ZMod (3 ^ T)) : Prop := ∃ eta : ZMod (3 ^ T), omega = (3 : ZMod (3 ^ T)) ^ q * eta /-- Divisibility by `3^q` in `ZMod (3^T)` is detected by the canonical natural representative whenever `q ≤ T`. -/ theorem zmodThreePowMultiple_iff_dvd_val {T q : ℕ} (hq : q ≤ T) (omega : ZMod (3 ^ T)) : zmodThreePowMultiple T q omega ↔ 3 ^ q ∣ omega.val := by constructor · rintro ⟨eta, homega⟩ have hcast : (omega.val : ZMod (3 ^ T)) = ((3 ^ q * eta.val : ℕ) : ZMod (3 ^ T)) := by calc (omega.val : ZMod (3 ^ T)) = omega := ZMod.natCast_zmod_val omega _ = (3 : ZMod (3 ^ T)) ^ q * eta := homega _ = (3 : ZMod (3 ^ T)) ^ q * (eta.val : ZMod (3 ^ T)) := by rw [ZMod.natCast_zmod_val] _ = ((3 ^ q * eta.val : ℕ) : ZMod (3 ^ T)) := by simp only [Nat.cast_mul, Nat.cast_pow] norm_num have hmod : omega.val ≡ 3 ^ q * eta.val [MOD 3 ^ T] := (ZMod.natCast_eq_natCast_iff omega.val (3 ^ q * eta.val) (3 ^ T)).mp hcast exact (hmod.dvd_iff (Nat.pow_dvd_pow 3 hq)).mpr (dvd_mul_right (3 ^ q) eta.val) · rintro ⟨eta, heta⟩ refine ⟨(eta : ZMod (3 ^ T)), ?_⟩ rw [← ZMod.natCast_zmod_val omega, heta] push_cast rfl /-- Multiplication by a unit preserves every power-of-three frequency subgroup. -/ theorem zmodThreePowMultiple_unit_mul_iff {T q : ℕ} (u : (ZMod (3 ^ T))ˣ) (omega : ZMod (3 ^ T)) : zmodThreePowMultiple T q ((u : ZMod (3 ^ T)) * omega) ↔ zmodThreePowMultiple T q omega := by constructor · rintro ⟨eta, heta⟩ refine ⟨(u⁻¹ : ZMod (3 ^ T)) * eta, ?_⟩ calc omega = (u⁻¹ : ZMod (3 ^ T)) * ((u : ZMod (3 ^ T)) * omega) := by simp _ = (u⁻¹ : ZMod (3 ^ T)) * ((3 : ZMod (3 ^ T)) ^ q * eta) := by rw [heta] _ = (3 : ZMod (3 ^ T)) ^ q * ((u⁻¹ : ZMod (3 ^ T)) * eta) := by ring · rintro ⟨eta, heta⟩ refine ⟨(u : ZMod (3 ^ T)) * eta, ?_⟩ rw [heta] ring theorem taoZModThreeProjection_eq_natCast_val {r T : ℕ} (h : r ≤ T) (omega : ZMod (3 ^ T)) : taoZModThreeProjection h omega = (omega.val : ZMod (3 ^ r)) := by conv_lhs => rw [← ZMod.natCast_zmod_val omega] exact taoZModThreeProjection_natCast h omega.val /-- Projection from level `T` to level `r` preserves divisibility by `3^q` when `q ≤ r`. -/ theorem zmodThreePowMultiple_projection_iff {q r T : ℕ} (hqr : q ≤ r) (hrT : r ≤ T) (omega : ZMod (3 ^ T)) : zmodThreePowMultiple T q omega ↔ zmodThreePowMultiple r q (taoZModThreeProjection hrT omega) := by rw [zmodThreePowMultiple_iff_dvd_val (hqr.trans hrT)] rw [zmodThreePowMultiple_iff_dvd_val hqr] have hcast : (omega.val : ZMod (3 ^ r)) = ((taoZModThreeProjection hrT omega).val : ZMod (3 ^ r)) := by rw [← taoZModThreeProjection_eq_natCast_val hrT omega] rw [ZMod.natCast_zmod_val] have hmod : omega.val ≡ (taoZModThreeProjection hrT omega).val [MOD 3 ^ r] := (ZMod.natCast_eq_natCast_iff omega.val (taoZModThreeProjection hrT omega).val (3 ^ r)).mp hcast exact hmod.dvd_iff (Nat.pow_dvd_pow 3 hqr) /-- A natural number not divisible by `3` gives a primitive class at every positive power-of-three level. -/ theorem zmodThreePrimitive_natCast_of_not_dvd {r eta : ℕ} (hr : 1 ≤ r) (heta : ¬3 ∣ eta) : zmodThreePrimitive r (eta : ZMod (3 ^ r)) := by intro hmultiple rcases hmultiple with ⟨zeta, hzeta⟩ have hcast : (eta : ZMod (3 ^ r)) = ((3 * zeta.val : ℕ) : ZMod (3 ^ r)) := by calc (eta : ZMod (3 ^ r)) = 3 * zeta := hzeta _ = 3 * (zeta.val : ZMod (3 ^ r)) := by rw [ZMod.natCast_zmod_val] _ = ((3 * zeta.val : ℕ) : ZMod (3 ^ r)) := by simp only [Nat.cast_mul] norm_num have hmod : eta ≡ 3 * zeta.val [MOD 3 ^ r] := (ZMod.natCast_eq_natCast_iff eta (3 * zeta.val) (3 ^ r)).mp hcast apply heta exact (hmod.dvd_iff (by refine ⟨3 ^ (r - 1), ?_⟩ rw [← pow_succ'] congr omega)).mpr (dvd_mul_right 3 zeta.val) /-- Additive-level adapter for a natural primitive cofactor. -/ theorem syracPMFPrimitivePolynomialDecayAt.dft_three_pow_mul_natCast_le_of_add_eq {A : ℕ} {C : ℝ} (hdecay : syracPMFPrimitivePolynomialDecayAt A C) {r j T eta : ℕ} (hadd : r + j = T) (hr : 1 ≤ r) (heta : ¬3 ∣ eta) : ‖ZMod.dft (pmfComplexMass (syracPMF T)) ((3 : ZMod (3 ^ T)) ^ j * eta)‖ ≤ C / (r : ℝ) ^ A := by subst T apply hdecay.dft_three_pow_mul_le hr simpa using zmodThreePrimitive_natCast_of_not_dvd hr heta /-- A frequency outside the `3^q` subgroup has an exact factorization `3^j * eta` with `j < q` and `eta` not divisible by `3`. -/ theorem exists_three_pow_mul_not_dvd_of_not_pow_multiple {T q : ℕ} {omega : ZMod (3 ^ T)} (homega : ¬zmodThreePowMultiple T q omega) : ∃ j eta : ℕ, j < q ∧ ¬3 ∣ eta ∧ omega = (3 : ZMod (3 ^ T)) ^ j * eta := by have homega_ne : omega ≠ 0 := by intro hz apply homega exact ⟨0, by simp [hz]⟩ have hval_ne : omega.val ≠ 0 := (ZMod.val_ne_zero omega).mpr homega_ne obtain ⟨j, eta, heta, hval⟩ := Nat.exists_eq_pow_mul_and_not_dvd hval_ne 3 (by omega) have homega_factor : omega = (3 : ZMod (3 ^ T)) ^ j * eta := by calc omega = (omega.val : ZMod (3 ^ T)) := (ZMod.natCast_zmod_val omega).symm _ = ((3 ^ j * eta : ℕ) : ZMod (3 ^ T)) := by rw [hval] _ = (3 : ZMod (3 ^ T)) ^ j * eta := by simp only [Nat.cast_mul, Nat.cast_pow] norm_num have hj : j < q := by by_contra hj_not have hqj : q ≤ j := Nat.le_of_not_gt hj_not apply homega refine ⟨(3 : ZMod (3 ^ T)) ^ (j - q) * eta, ?_⟩ calc omega = (3 : ZMod (3 ^ T)) ^ j * eta := homega_factor _ = ((3 : ZMod (3 ^ T)) ^ q * (3 : ZMod (3 ^ T)) ^ (j - q)) * eta := by rw [← pow_add, Nat.add_sub_of_le hqj] _ = (3 : ZMod (3 ^ T)) ^ q * ((3 : ZMod (3 ^ T)) ^ (j - q) * eta) := by ring exact ⟨j, eta, hj, heta, homega_factor⟩ /-- Tao's tail frequency after projection and multiplication by `2^{-l}`. -/ noncomputable def taoSection6TailScaledFrequency {T n : ℕ} (hTn : T ≤ n) (l : ℕ) (xi : ZMod (3 ^ n)) : ZMod (3 ^ T) := ((taoCor63TwoPowUnit T l)⁻¹ : ZMod (3 ^ T)) * taoZModThreeProjection hTn xi /-- Projection to the tail followed by `2^{-l}` scaling preserves exclusion from the `3^q` frequency subgroup. -/ theorem taoSection6TailScaledFrequency_not_pow_multiple {q T n l : ℕ} (hqT : q ≤ T) (hTn : T ≤ n) {xi : ZMod (3 ^ n)} (hxi : ¬zmodThreePowMultiple n q xi) : ¬zmodThreePowMultiple T q (taoSection6TailScaledFrequency hTn l xi) := by intro hscaled apply hxi apply (zmodThreePowMultiple_projection_iff hqT hTn xi).mpr apply (zmodThreePowMultiple_unit_mul_iff ((taoCor63TwoPowUnit T l)⁻¹) (taoZModThreeProjection hTn xi)).mp simpa [taoSection6TailScaledFrequency] using hscaled /-- Source-facing stopped-tail conductor packet. Every discarded ambient frequency yields a positive reduced conductor `r`, a factorization exponent `j`, a quantitative comparison `n ≤ 20r`, and the corresponding primitive Fourier-decay bound for the `2^{-l}`-scaled tail frequency. -/ theorem syracPMFPrimitivePolynomialDecayAt.tail_scaled_frequency_le {A : ℕ} {C : ℝ} (hdecay : syracPMFPrimitivePolynomialDecayAt A C) {n m k T l : ℕ} (htail : k + 1 + T = n) (hhead : k + 1 ≤ m) (hm : 9 * n ≤ 10 * m) (hk : 20 * k ≤ 17 * n) {xi : ZMod (3 ^ n)} (hxi : ¬zmodThreePowMultiple n (n - m) xi) : ∃ r j : ℕ, j < n - m ∧ r + j = T ∧ k + j + 1 + r = n ∧ 1 ≤ r ∧ m - k ≤ r ∧ n ≤ 20 * r ∧ ‖ZMod.dft (pmfComplexMass (syracPMF T)) (taoSection6TailScaledFrequency (by omega : T ≤ n) l xi)‖ ≤ C / (r : ℝ) ^ A := by have hqT : n - m ≤ T := by omega have hTn : T ≤ n := by omega have hscaled : ¬zmodThreePowMultiple T (n - m) (taoSection6TailScaledFrequency hTn l xi) := taoSection6TailScaledFrequency_not_pow_multiple hqT hTn hxi obtain ⟨j, eta, hj, heta, hfactor⟩ := exists_three_pow_mul_not_dvd_of_not_pow_multiple hscaled have hjT : j < T := hj.trans_le hqT let r := T - j have hjr : j + r = T := by dsimp [r] omega have hbounds := taoSection6_conductor_index_bounds htail hjr hhead hj hm hk rcases hbounds with ⟨hrj, hr, hmkr, _hnmk, hnr⟩ have hfull := taoSection6_head_add_frequency_add_conductor_eq htail hjr have hdecayTail : ‖ZMod.dft (pmfComplexMass (syracPMF T)) ((3 : ZMod (3 ^ T)) ^ j * eta)‖ ≤ C / (r : ℝ) ^ A := by exact hdecay.dft_three_pow_mul_natCast_le_of_add_eq hrj hr heta have htailDecay : ‖ZMod.dft (pmfComplexMass (syracPMF T)) (taoSection6TailScaledFrequency hTn l xi)‖ ≤ C / (r : ℝ) ^ A := by rw [hfactor] exact hdecayTail exact ⟨r, j, hj, hrj, hfull, hr, hmkr, hnr, htailDecay⟩ end end Tao end Erdos1135