import Erdos1135.Tao.Fourier.DecayStatement import Erdos1135.Tao.Fourier.Lemma74ThetaArithmetic /-! # Lemma 7.4 Primitive Strip This proof leaf supplies the primitive-frequency arithmetic used to keep Section 7 black points away from the right boundary and to find a bounded white point above them. It stops before canonical selectors and triangle families. -/ namespace Erdos1135 namespace Tao noncomputable section /-- A primitive class modulo `3^n` is a unit. This uses nondivisibility by `3`, not merely nonzeroness. -/ theorem zmodThreePrimitive.isUnit {n : ℕ} {xi : ZMod (3 ^ n)} (hxi : zmodThreePrimitive n xi) : IsUnit xi := by have hthree : ¬ 3 ∣ xi.val := by rintro ⟨k, hk⟩ apply hxi refine ⟨(k : ZMod (3 ^ n)), ?_⟩ rw [← ZMod.natCast_zmod_val xi, hk, Nat.cast_mul] norm_num rw [← ZMod.natCast_zmod_val xi] exact (ZMod.isUnit_iff_coprime xi.val (3 ^ n)).2 (Nat.prime_three.coprime_pow_of_not_dvd hthree) /-- Multiplication by `3^(n-1)` sends every unit modulo `3^n` to one of the two nonzero elements annihilated by `3`, hence its centered numerator has magnitude `3^(n-1)`. -/ theorem natAbs_valMinAbs_three_pow_pred_mul_unit {n : ℕ} (hn : 1 ≤ n) {u : ZMod (3 ^ n)} (hu : IsUnit u) : (((3 : ZMod (3 ^ n)) ^ (n - 1) * u).valMinAbs).natAbs = 3 ^ (n - 1) := by let p : ℕ := 3 ^ (n - 1) let x : ZMod (3 ^ n) := (p : ZMod (3 ^ n)) * u have hp : 0 < p := by dsimp [p] positivity have hfactor : p * 3 = 3 ^ n := by dsimp [p] exact pow_sub_one_mul (by omega) 3 have hp_lt : p < 3 ^ n := by rw [← hfactor] omega have hp_cast_ne : (p : ZMod (3 ^ n)) ≠ 0 := by intro hzero exact (Nat.not_dvd_of_pos_of_lt hp hp_lt) ((ZMod.natCast_eq_zero_iff p (3 ^ n)).1 hzero) have hx_ne : x ≠ 0 := by intro hx apply hp_cast_ne apply hu.mul_left_injective simpa [x] using hx have hthree_x : (3 : ZMod (3 ^ n)) * x = 0 := by dsimp [x] calc (3 : ZMod (3 ^ n)) * ((p : ZMod (3 ^ n)) * u) = ((p * 3 : ℕ) : ZMod (3 ^ n)) * u := by push_cast ring _ = 0 := by simp [hfactor] have hcast : (((3 : ℤ) * x.valMinAbs : ℤ) : ZMod (3 ^ n)) = 0 := by rw [Int.cast_mul, ZMod.coe_valMinAbs] simpa using hthree_x have hbig : ((3 ^ n : ℕ) : ℤ) ∣ (3 : ℤ) * x.valMinAbs := (ZMod.intCast_zmod_eq_zero_iff_dvd _ _).1 hcast have hsmall : (p : ℤ) ∣ x.valMinAbs := by have hfactorInt : (3 : ℤ) * (p : ℤ) = ((3 ^ n : ℕ) : ℤ) := by exact_mod_cast (by simpa [mul_comm] using hfactor) apply Int.dvd_of_mul_dvd_mul_left (by norm_num : (3 : ℤ) ≠ 0) rwa [hfactorInt] have hsmallNat : p ∣ x.valMinAbs.natAbs := Int.natCast_dvd.mp hsmall have hnum_ne : x.valMinAbs.natAbs ≠ 0 := by simpa [Int.natAbs_eq_zero, ZMod.valMinAbs_eq_zero] using hx_ne have hupper : x.valMinAbs.natAbs ≤ (3 ^ n) / 2 := ZMod.natAbs_valMinAbs_le x have hhalf : (3 ^ n) / 2 < 2 * p := by rw [← hfactor] omega have hlt : x.valMinAbs.natAbs < 2 * p := hupper.trans_lt hhalf rcases hsmallNat with ⟨q, hq⟩ have hq_pos : 0 < q := by apply Nat.pos_of_ne_zero intro hq_zero apply hnum_ne simpa [hq_zero] using hq have hq_lt : q < 2 := by apply (Nat.mul_lt_mul_left hp).mp simpa [hq, mul_comm] using hlt have hq_one : q = 1 := by omega subst q simpa [x, p] using hq /-- Tao's terminal primitive residue has exact centered magnitude `1/3`. -/ theorem abs_taoSignedTheta_three_pow_pred_mul_unit {n : ℕ} (hn : 1 ≤ n) {u : ZMod (3 ^ n)} (hu : IsUnit u) : |taoSignedTheta ((3 : ZMod (3 ^ n)) ^ (n - 1) * u)| = (1 : ℝ) / 3 := by let x : ZMod (3 ^ n) := (3 : ZMod (3 ^ n)) ^ (n - 1) * u have hnumNat : x.valMinAbs.natAbs = 3 ^ (n - 1) := by simpa [x] using natAbs_valMinAbs_three_pow_pred_mul_unit hn hu have hnumReal : |((x.valMinAbs : ℤ) : ℝ)| = ((3 ^ (n - 1) : ℕ) : ℝ) := by rw [← Int.cast_abs, Int.abs_eq_natAbs] exact_mod_cast hnumNat have hfactor : ((3 ^ n : ℕ) : ℝ) = 3 * ((3 ^ (n - 1) : ℕ) : ℝ) := by have hfactorNat : 3 ^ n = 3 * 3 ^ (n - 1) := by calc 3 ^ n = 3 ^ (n - 1) * 3 := (pow_sub_one_mul (by omega : n ≠ 0) 3).symm _ = 3 * 3 ^ (n - 1) := by ring exact_mod_cast hfactorNat have hpReal : (0 : ℝ) < ((3 ^ (n - 1) : ℕ) : ℝ) := by positivity rw [taoSignedTheta, abs_div, abs_of_pos (by positivity : (0 : ℝ) < (3 ^ n : ℕ))] change |((x.valMinAbs : ℤ) : ℝ)| / ((3 ^ n : ℕ) : ℝ) = _ rw [hnumReal, hfactor] field_simp /-- Multiplying a primitive class by an integral power of `2` preserves unitness. -/ theorem taoSection7TwoZPow_mul_primitive_isUnit {n : ℕ} {xi : ZMod (3 ^ n)} (hxi : zmodThreePrimitive n xi) (l : ℤ) : IsUnit (taoSection7TwoZPow n (1 - l) * xi) := by have htwo : IsUnit (taoSection7TwoZPow n (1 - l)) := by unfold taoSection7TwoZPow exact Units.isUnit _ exact htwo.mul hxi.isUnit /-- Exact modular amplification of a Section 7 theta residue to its terminal `3`-adic scale. -/ theorem three_pow_amplify_taoSection7ThetaResidue {n : ℕ} (xi : ZMod (3 ^ n)) (j : ℕ+) (l : ℤ) (hj : (j : ℕ) ≤ n / 2) : (3 : ZMod (3 ^ n)) ^ (n + 1 - 2 * (j : ℕ)) * taoSection7ThetaResidue n xi j l = (3 : ZMod (3 ^ n)) ^ (n - 1) * (taoSection7TwoZPow n (1 - l) * xi) := by have hexp : (n + 1 - 2 * (j : ℕ)) + 2 * ((j : ℕ) - 1) = n - 1 := by have hjpos : 0 < (j : ℕ) := j.pos omega unfold taoSection7ThetaResidue calc (3 : ZMod (3 ^ n)) ^ (n + 1 - 2 * (j : ℕ)) * (((3 : ZMod (3 ^ n)) ^ (2 * ((j : ℕ) - 1)) * taoSection7TwoZPow n (1 - l)) * xi) = (3 : ZMod (3 ^ n)) ^ ((n + 1 - 2 * (j : ℕ)) + 2 * ((j : ℕ) - 1)) * (taoSection7TwoZPow n (1 - l) * xi) := by rw [pow_add] ring _ = _ := by rw [hexp] /-- Primitive theta residues in the left half-strip cannot be too small after amplification to the terminal `3`-adic scale. -/ theorem one_third_le_three_pow_mul_abs_taoSection7ThetaResidue {n : ℕ} {xi : ZMod (3 ^ n)} (hxi : zmodThreePrimitive n xi) (j : ℕ+) (hj : (j : ℕ) ≤ n / 2) (l : ℤ) : (1 : ℝ) / 3 ≤ (3 : ℝ) ^ (n + 1 - 2 * (j : ℕ)) * |taoSignedTheta (taoSection7ThetaResidue n xi j l)| := by have hn : 1 ≤ n := by have hjpos : 0 < (j : ℕ) := j.pos omega have hu := taoSection7TwoZPow_mul_primitive_isUnit hxi l have hterminal := abs_taoSignedTheta_three_pow_pred_mul_unit hn hu have hscale := abs_taoSignedTheta_natCast_mul_le (N := 3 ^ n) (3 ^ (n + 1 - 2 * (j : ℕ))) (taoSection7ThetaResidue n xi j l) have hid := three_pow_amplify_taoSection7ThetaResidue xi j l hj have hcastIdentity : ((3 ^ (n + 1 - 2 * (j : ℕ)) : ℕ) : ZMod (3 ^ n)) * taoSection7ThetaResidue n xi j l = (3 : ZMod (3 ^ n)) ^ (n - 1) * (taoSection7TwoZPow n (1 - l) * xi) := by simpa using hid rw [hcastIdentity, hterminal] at hscale simpa using hscale /-- A black primitive point obeys Tao's raw amplified strip inequality. -/ theorem one_third_le_three_pow_mul_epsilon_of_sourceBlackPoint {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hxi : zmodThreePrimitive n xi) (p : TaoSection7Point) (hp : (p.j : ℕ) ≤ n / 2) (hblack : taoSection7SourceBlackPoint n xi epsilon p) : (1 : ℝ) / 3 ≤ (3 : ℝ) ^ (n + 1 - 2 * (p.j : ℕ)) * epsilon := by calc (1 : ℝ) / 3 ≤ (3 : ℝ) ^ (n + 1 - 2 * (p.j : ℕ)) * |taoSection7SourceTheta n xi p| := by exact one_third_le_three_pow_mul_abs_taoSection7ThetaResidue hxi p.j hp p.l _ ≤ (3 : ℝ) ^ (n + 1 - 2 * (p.j : ℕ)) * epsilon := by exact mul_le_mul_of_nonneg_left hblack (by positivity) /-- Logarithmic coordinate form of the primitive black-strip inequality. -/ theorem taoSection7SourceBlackPoint_raw_log_coordinate {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hxi : zmodThreePrimitive n xi) (hepsilon : 0 < epsilon) (p : TaoSection7Point) (hp : (p.j : ℕ) ≤ n / 2) (hblack : taoSection7SourceBlackPoint n xi epsilon p) : 2 * (((p.j : ℕ) : ℝ)) + Real.log (1 / (3 * epsilon)) / Real.log 3 ≤ (n : ℝ) + 1 := by have hamp := one_third_le_three_pow_mul_epsilon_of_sourceBlackPoint hxi p hp hblack have hthreeEpsilon : 0 < (3 : ℝ) * epsilon := mul_pos (by norm_num) hepsilon have hdiv : (1 : ℝ) / (3 * epsilon) ≤ (3 : ℝ) ^ (n + 1 - 2 * (p.j : ℕ)) := by rw [div_le_iff₀ hthreeEpsilon] nlinarith have hlog := Real.log_le_log (div_pos (by norm_num) hthreeEpsilon) hdiv rw [Real.log_pow] at hlog have hlog3 : 0 < Real.log (3 : ℝ) := Real.log_pos (by norm_num) have hquotient : Real.log (1 / (3 * epsilon)) / Real.log 3 ≤ ((n + 1 - 2 * (p.j : ℕ) : ℕ) : ℝ) := by rw [div_le_iff₀ hlog3] simpa [mul_comm] using hlog have hsub : 2 * (p.j : ℕ) ≤ n + 1 := by omega have hcast : ((n + 1 - 2 * (p.j : ℕ) : ℕ) : ℝ) = (n : ℝ) + 1 - 2 * (((p.j : ℕ) : ℝ)) := by rw [Nat.cast_sub hsub] push_cast ring rw [hcast] at hquotient linarith /-- The source regime `epsilon <= 1/100` supplies the scalar margin used to turn the raw logarithmic coordinate bound into Tao's right-strip inset. -/ theorem one_add_taoSection7TriangleSeparation_le_logScale_div_log9 {epsilon : ℝ} (hepsilon0 : 0 < epsilon) (hepsilon100 : epsilon ≤ (1 / 100 : ℝ)) : 1 + taoSection7TriangleSeparation epsilon ≤ taoSection7TriangleLogScale epsilon / Real.log 9 := by let L : ℝ := taoSection7TriangleLogScale epsilon have hlog9 : 0 < Real.log (9 : ℝ) := Real.log_pos (by norm_num) have hhundred : (100 : ℝ) ≤ 1 / epsilon := by rw [le_div_iff₀ hepsilon0] nlinarith have heightyOne : (81 : ℝ) ≤ 1 / epsilon := by linarith have htwoLog9 : 2 * Real.log 9 ≤ L := by have hlog := Real.log_le_log (by norm_num : (0 : ℝ) < 81) heightyOne rw [show (81 : ℝ) = 9 ^ 2 by norm_num, Real.log_pow] at hlog simpa [L, taoSection7TriangleLogScale] using hlog have hlog3_le_two : Real.log (3 : ℝ) ≤ 2 := by have h := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 3) norm_num at h exact h have hlog9_eq : Real.log (9 : ℝ) = 2 * Real.log 3 := by rw [show (9 : ℝ) = 3 ^ 2 by norm_num, Real.log_pow] norm_num have hlog9_le_five : Real.log (9 : ℝ) ≤ 5 := by rw [hlog9_eq] linarith have hL0 : 0 ≤ L := by linarith have hfirstHalf : Real.log 9 ≤ L / 2 := by linarith have hsecondHalf : L * Real.log 9 / 10 ≤ L / 2 := by have hproduct : 0 ≤ L * (5 - Real.log 9) := mul_nonneg hL0 (sub_nonneg.mpr hlog9_le_five) nlinarith apply (le_div_iff₀ hlog9).2 change (1 + (1 / 10 : ℝ) * L) * Real.log 9 ≤ L nlinarith /-- Primitive black points in the source regime lie inside Tao's real right-strip boundary. The conclusion retains the real half `(n : ℝ) / 2`, not the floored natural cutoff. -/ theorem taoSection7SourceBlackPoint_jReal_le_triangleRightBound {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hxi : zmodThreePrimitive n xi) (hepsilon0 : 0 < epsilon) (hepsilon100 : epsilon ≤ (1 / 100 : ℝ)) (p : TaoSection7Point) (hp : (p.j : ℕ) ≤ n / 2) (hblack : taoSection7SourceBlackPoint n xi epsilon p) : (((p.j : ℕ) : ℝ)) ≤ taoSection7TriangleRightBound n epsilon := by let L : ℝ := taoSection7TriangleLogScale epsilon let rho : ℝ := taoSection7TriangleSeparation epsilon have hraw := taoSection7SourceBlackPoint_raw_log_coordinate hxi hepsilon0 p hp hblack have hscalar := one_add_taoSection7TriangleSeparation_le_logScale_div_log9 hepsilon0 hepsilon100 have hlog3 : 0 < Real.log (3 : ℝ) := Real.log_pos (by norm_num) have hlog9Eq : Real.log (9 : ℝ) = 2 * Real.log 3 := by rw [show (9 : ℝ) = 3 ^ 2 by norm_num, Real.log_pow] norm_num have harg : (1 : ℝ) / (3 * epsilon) = (1 / epsilon) / 3 := by field_simp have hlogArg : Real.log (1 / (3 * epsilon)) = L - Real.log 3 := by rw [harg, Real.log_div (one_div_ne_zero hepsilon0.ne') (by norm_num)] rfl have hratio : Real.log (1 / (3 * epsilon)) / Real.log 3 = 2 * (L / Real.log 9) - 1 := by rw [hlogArg, hlog9Eq] field_simp [hlog3.ne'] have hmargin : 1 + 2 * rho ≤ Real.log (1 / (3 * epsilon)) / Real.log 3 := by change 1 + rho ≤ L / Real.log 9 at hscalar rw [hratio] linarith unfold taoSection7TriangleRightBound change (((p.j : ℕ) : ℝ)) ≤ (n : ℝ) / 2 - rho linarith namespace TaoSection7Point /-- The point `t` integral steps above `p`. -/ def upN (p : TaoSection7Point) (t : ℕ) : TaoSection7Point := ⟨p.j, p.l + t⟩ @[simp] theorem upN_zero (p : TaoSection7Point) : p.upN 0 = p := by cases p simp [upN] @[simp] theorem upN_j (p : TaoSection7Point) (t : ℕ) : (p.upN t).j = p.j := by rfl @[simp] theorem upN_l (p : TaoSection7Point) (t : ℕ) : (p.upN t).l = p.l + t := by rfl @[simp] theorem upN_succ_down (p : TaoSection7Point) (t : ℕ) : (p.upN (t + 1)).down = p.upN t := by cases p simp [upN, down] omega end TaoSection7Point /-- A black vertical prefix transports the upper theta back to its starting value by an exact power of `2`. -/ theorem pow_two_mul_sourceTheta_upN_eq_of_black_prefix {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (p : TaoSection7Point) (t : ℕ) (hepsilon : epsilon < 1 / 100) (hblack : ∀ u, u ≤ t → taoSection7SourceBlackPoint n xi epsilon (p.upN u)) : (2 : ℝ) ^ t * taoSection7SourceTheta n xi (p.upN t) = taoSection7SourceTheta n xi p := by induction t with | zero => simp | succ t ih => have hupper := hblack (t + 1) (by omega) have hguard : |2 * taoSection7SourceTheta n xi (p.upN (t + 1))| < (1 / 2 : ℝ) := by calc |2 * taoSection7SourceTheta n xi (p.upN (t + 1))| = 2 * |taoSection7SourceTheta n xi (p.upN (t + 1))| := by rw [abs_mul] norm_num _ ≤ 2 * epsilon := mul_le_mul_of_nonneg_left hupper (by norm_num) _ < 1 / 2 := by linarith have hdown := taoSection7SourceTheta_down_of_abs_lt_half n xi (p.upN (t + 1)) hguard rw [TaoSection7Point.upN_succ_down] at hdown have ih' := ih (fun u hu => hblack u (by omega)) rw [pow_succ] calc (2 : ℝ) ^ t * 2 * taoSection7SourceTheta n xi (p.upN (t + 1)) = (2 : ℝ) ^ t * taoSection7SourceTheta n xi (p.upN t) := by rw [hdown] ring _ = taoSection7SourceTheta n xi p := ih' /-- Every primitive black point in the left half-strip has a white point at one of the next `2*n` vertical offsets. -/ theorem exists_sourceWhitePoint_upN_le_two_mul {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} (hn : 1 ≤ n) (hxi : zmodThreePrimitive n xi) (p : TaoSection7Point) (hp : (p.j : ℕ) ≤ n / 2) (hstart : taoSection7SourceBlackPoint n xi epsilon p) (hepsilon : epsilon < 1 / 100) : ∃ t : ℕ, 1 ≤ t ∧ t ≤ 2 * n ∧ taoSection7SourceWhitePoint n xi epsilon (p.upN t) := by by_contra hnone have hall : ∀ u, u ≤ 2 * n → taoSection7SourceBlackPoint n xi epsilon (p.upN u) := by intro u hu by_cases hu0 : u = 0 · simpa [hu0] using hstart · have hnotWhite : ¬ taoSection7SourceWhitePoint n xi epsilon (p.upN u) := by intro hwhite apply hnone exact ⟨u, Nat.one_le_iff_ne_zero.2 hu0, hu, hwhite⟩ simpa [taoSection7SourceWhitePoint, taoSection7White, taoSection7SourceBlackPoint, taoSection7Black, not_lt] using hnotWhite have htransport := pow_two_mul_sourceTheta_upN_eq_of_black_prefix p (2 * n) hepsilon hall have hpowTwo : (2 : ℝ) ^ (2 * n) = (4 : ℝ) ^ n := by calc (2 : ℝ) ^ (2 * n) = ((2 : ℝ) ^ 2) ^ n := by rw [pow_mul] _ = (4 : ℝ) ^ n := by norm_num have hdecay : (4 : ℝ) ^ n * |taoSection7SourceTheta n xi (p.upN (2 * n))| ≤ epsilon := by calc (4 : ℝ) ^ n * |taoSection7SourceTheta n xi (p.upN (2 * n))| = |(2 : ℝ) ^ (2 * n) * taoSection7SourceTheta n xi (p.upN (2 * n))| := by rw [abs_mul, abs_pow, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2), hpowTwo] _ = |taoSection7SourceTheta n xi p| := by rw [htransport] _ ≤ epsilon := hstart have hamp := one_third_le_three_pow_mul_abs_taoSection7ThetaResidue hxi p.j hp (p.upN (2 * n)).l have hamp' : (1 : ℝ) / 3 ≤ (3 : ℝ) ^ (n + 1 - 2 * (p.j : ℕ)) * |taoSection7SourceTheta n xi (p.upN (2 * n))| := by simpa [taoSection7SourceTheta] using hamp have hexp : n + 1 - 2 * (p.j : ℕ) ≤ n - 1 := by have hjpos : 0 < (p.j : ℕ) := p.j.pos omega have hpowBound : (3 : ℝ) ^ (n + 1 - 2 * (p.j : ℕ)) ≤ (4 : ℝ) ^ n := by calc (3 : ℝ) ^ (n + 1 - 2 * (p.j : ℕ)) ≤ (3 : ℝ) ^ (n - 1) := pow_le_pow_right₀ (by norm_num) hexp _ ≤ (4 : ℝ) ^ (n - 1) := by gcongr norm_num _ ≤ (4 : ℝ) ^ n := by apply pow_le_pow_right₀ (by norm_num) omega have hscaled : (3 : ℝ) ^ (n + 1 - 2 * (p.j : ℕ)) * |taoSection7SourceTheta n xi (p.upN (2 * n))| ≤ (4 : ℝ) ^ n * |taoSection7SourceTheta n xi (p.upN (2 * n))| := mul_le_mul_of_nonneg_right hpowBound (abs_nonneg _) have hcontra : (1 : ℝ) / 3 < 1 / 3 := hamp'.trans (hscaled.trans hdecay) |>.trans_lt (hepsilon.trans (by norm_num)) exact (lt_irrefl ((1 : ℝ) / 3)) hcontra end end Tao end Erdos1135