| import Mathlib |
| import OAI.NumberTheory.CubicGram.GramDyad |
|
|
| namespace OAI |
|
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| /-! Scale-normalized coprime envelopes and divisor losses. -/ |
|
|
| section |
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| noncomputable section |
| open scoped BigOperators ContDiff |
| open Set Filter MeasureTheory Topology |
| attribute [local instance] Classical.propDecidable |
| namespace CubicFirstMoment |
|
|
| lemma gram_coprime_envelope {N a Z ε : ℝ} (hN : 0 < N) (ha : 0 < a) |
| (haN : a ≤ 2*N) (hZ : 0 < Z) (hε : 0 < ε) : |
| let c := Z/(27*a*N) |
| (Z^2/(81*a*N))*(((2*N)^ε/c)*((2*N)*gramDyadConstant ε*c^(-ε) + |
| gramDyadConstant (1+ε)*c^(-(1+ε)) + |
| (2*N)^(2/3 : ℝ)*gramDyadConstant (2/3+ε)*c^(-(2/3+ε)))) ≤ |
| (108*(gramDyadConstant ε+gramDyadConstant (1+ε)+gramDyadConstant (2/3+ε))*(108 : ℝ)^ε) * |
| (N^3/Z)^ε*(Z*N+N^2+N^2*Z^(1/3 : ℝ)) := by |
| let q : ℝ := 27*a*N/Z |
| have hq : 0 < q := by dsimp [q]; positivity |
| have hi : Z/(27*a*N) = q⁻¹ := by dsimp [q]; rw [inv_div] |
| dsimp only |
| rw [hi] |
| have hp (p : ℝ) : (q⁻¹)^(-p) = q^p := by |
| rw [Real.inv_rpow hq.le,Real.rpow_neg hq.le,inv_inv] |
| rw [hp,hp,hp,Real.rpow_add hq,Real.rpow_add hq,Real.rpow_one] |
| let D := gramDyadConstant ε+gramDyadConstant (1+ε)+gramDyadConstant (2/3+ε) |
| have hk0 := gramDyadConstant_pos hε |
| have hk1 := gramDyadConstant_pos (by linarith : 0 < 1+ε) |
| have hk2 := gramDyadConstant_pos (by linarith : 0 < 2/3+ε) |
| have hD : 0 < D := by dsimp [D]; positivity |
| have he : Z^2/(81*a*N)*((2*N)^ε/q⁻¹*(2*N*gramDyadConstant ε*q^ε + |
| gramDyadConstant (1+ε)*(q*q^ε)+(2*N)^(2/3 : ℝ)*gramDyadConstant (2/3+ε)*(q^(2/3 : ℝ)*q^ε))) = |
| (1/3 : ℝ)*(2*N*q)^ε*(2*N*gramDyadConstant ε*Z+ |
| gramDyadConstant (1+ε)*(Z*q)+gramDyadConstant (2/3+ε)*Z*(2*N*q)^(2/3 : ℝ)) := by |
| rw [Real.mul_rpow (by positivity : 0 ≤ 2*N) hq.le, |
| Real.mul_rpow (by positivity : 0 ≤ 2*N) hq.le] |
| dsimp [q] |
| field_simp |
| ring |
| rw [he] |
| have hB : 2*N*q ≤ 108*(N^3/Z) := by |
| dsimp [q] |
| calc |
| 2*N*(27*a*N/Z) = (54*a*N^2)/Z := by ring |
| _ ≤ (108*N^3)/Z := div_le_div_of_nonneg_right (by |
| nlinarith [mul_nonneg (by linarith : 0 ≤ 2*N-a) (sq_nonneg N)]) hZ.le |
| _ = _ := by ring |
| have heps := Real.rpow_le_rpow (by positivity : 0 ≤ 2*N*q) hB hε.le |
| rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 108) (by positivity : 0 ≤ N^3/Z)] at heps |
| have hqbound : Z*q ≤ 54*N^2 := by dsimp [q]; field_simp; nlinarith |
| have hthird : Z*(2*N*q)^(2/3 : ℝ) ≤ 108*N^2*Z^(1/3 : ℝ) := by |
| have h := mul_le_mul_of_nonneg_left |
| (Real.rpow_le_rpow (by positivity : 0 ≤ 2*N*q) hB (by norm_num : (0 : ℝ) ≤ 2/3)) hZ.le |
| have hr : Z*(108*(N^3/Z))^(2/3 : ℝ) = (108 : ℝ)^(2/3 : ℝ)*N^2*Z^(1/3 : ℝ) := by |
| rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 108) (by positivity : 0 ≤ N^3/Z), |
| Real.div_rpow (by positivity : 0 ≤ N^3) hZ.le,← Real.rpow_natCast, |
| ← Real.rpow_mul hN.le] |
| norm_num |
| have heZ : Z/(Z^(2/3 : ℝ)) = Z^(1/3 : ℝ) := by |
| nth_rw 1 [← Real.rpow_one Z] |
| rw [← Real.rpow_sub hZ] |
| norm_num |
| rw [show Z*((108 : ℝ)^(2/3 : ℝ)*(N^2/Z^(2/3 : ℝ))) = |
| (108 : ℝ)^(2/3 : ℝ)*N^2*(Z/Z^(2/3 : ℝ)) by ring,heZ] |
| rw [hr] at h |
| apply h.trans |
| gcongr |
| exact (Real.rpow_le_rpow_of_exponent_le (by norm_num : (1 : ℝ) ≤ 108) |
| (by norm_num : (2/3 : ℝ) ≤ 1)).trans_eq (Real.rpow_one _) |
| have hb : 2*N*gramDyadConstant ε*Z+gramDyadConstant (1+ε)*(Z*q)+ |
| gramDyadConstant (2/3+ε)*Z*(2*N*q)^(2/3 : ℝ) ≤ |
| 108*D*(Z*N+N^2+N^2*Z^(1/3 : ℝ)) := by |
| have hk0D : gramDyadConstant ε ≤ D := by dsimp [D]; linarith |
| have hk1D : gramDyadConstant (1+ε) ≤ D := by dsimp [D]; linarith |
| have hk2D : gramDyadConstant (2/3+ε) ≤ D := by dsimp [D]; linarith |
| calc |
| _ ≤ D*(2*N*Z+(54*N^2)+(108*N^2*Z^(1/3 : ℝ))) := by |
| have h0 := mul_le_mul_of_nonneg_right hk0D (by positivity : 0 ≤ 2*N*Z) |
| have h1 := mul_le_mul hk1D hqbound (by positivity) hD.le |
| have h2 := mul_le_mul hk2D hthird (by positivity) hD.le |
| nlinarith |
| _ ≤ _ := by nlinarith [mul_nonneg (hD.le) (by positivity : 0 ≤ Z*N), |
| mul_nonneg hD.le (sq_nonneg N)] |
| have h := mul_le_mul heps hb (by positivity) (by positivity) |
| have hL : 0 ≤ (2*N*q)^ε*(2*N*gramDyadConstant ε*Z+gramDyadConstant (1+ε)*(Z*q)+ |
| gramDyadConstant (2/3+ε)*Z*(2*N*q)^(2/3 : ℝ)) := by positivity |
| dsimp [D] at h |
| nlinarith |
|
|
| theorem primaryCharacterGram_coprime_normalized (W : ℝ → ℂ) (hW : HasCompactSupport W) |
| (hW' : ContDiff ℝ ∞ W) {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (S : Finset Eisenstein) (N : ℝ), 1 ≤ 2*N → |
| (∀ b ∈ S, primary b ∧ Squarefree b ∧ N ≤ norm b ∧ norm b ≤ 2*N) → |
| ∀ (a : Eisenstein), primary a → Squarefree a → norm a ≤ 2*N → |
| (∀ b ∈ S, IsCoprime a b ∧ a ≠ b) → ∀ (Z : ℝ), 0 < Z → |
| (∑ b ∈ S, ‖primaryCharacterGram a b W Z‖^2) ≤ |
| C*(N^3/Z)^ε*(Z*N+N^2+N^2*Z^(1/3 : ℝ)) := by |
| obtain ⟨C,hC,hc⟩ := primaryCharacterGram_coprime_explicit W hW hW' hε hε1 |
| have hk0 := gramDyadConstant_pos hε |
| have hk1 := gramDyadConstant_pos (by linarith : 0 < 1+ε) |
| have hk2 := gramDyadConstant_pos (by linarith : 0 < 2/3+ε) |
| refine ⟨C*(108*(gramDyadConstant ε+gramDyadConstant (1+ε)+ |
| gramDyadConstant (2/3+ε))*(108 : ℝ)^ε),by positivity,?_⟩ |
| intro S N hN hS a ha hsa han hab Z hZ |
| have hNa : 0 < norm a := norm_pos_of_ne_zero (primary_ne_zero ha) |
| have h := mul_le_mul_of_nonneg_left (gram_coprime_envelope (by linarith) hNa han hZ hε) hC.le |
| have hh := hc S N hN hS a ha hsa hab Z hZ |
| dsimp only at h hh |
| nlinarith |
|
|
| lemma gram_scaled_loss {P Z K M ε : ℝ} (hP : 1 ≤ P) (hZ : 1 ≤ Z) |
| (hK : 1 ≤ K) (hM : 1 ≤ M) (hMP : M ≤ 2*P) (hε : 0 < ε) : |
| ((P/K)^3/(Z/M))^ε * ((Z/M)*(P/K)+(P/K)^2+(P/K)^2*(Z/M)^(1/3 : ℝ)) ≤ |
| (2 : ℝ)^ε * P^(4*ε) * (Z*P+P^2+P^2*Z^(1/3 : ℝ)) := by |
| have hP0 : 0 < P := by linarith |
| have hZ0 : 0 < Z := by linarith |
| have hK0 : 0 < K := by linarith |
| have hM0 : 0 < M := by linarith |
| have hnP : P/K ≤ P := div_le_self hP0.le hK |
| have hzZ : Z/M ≤ Z := div_le_self hZ0.le hM |
| have hx : (P/K)^3/(Z/M) ≤ 2*P^4 := by |
| rw [div_div_eq_mul_div] |
| calc |
| _ ≤ (P^3*(2*P))/1 := by gcongr |
| _ = _ := by ring |
| have hxpow := Real.rpow_le_rpow (by positivity : 0 ≤ (P/K)^3/(Z/M)) hx hε.le |
| rw [Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 2) (by positivity : 0 ≤ P^4), |
| ← Real.rpow_natCast P 4,← Real.rpow_mul hP0.le] at hxpow |
| have hshape : (Z/M)*(P/K)+(P/K)^2+(P/K)^2*(Z/M)^(1/3 : ℝ) ≤ |
| Z*P+P^2+P^2*Z^(1/3 : ℝ) := by gcongr |
| exact mul_le_mul hxpow hshape (by positivity) (by positivity) |
|
|
| end CubicFirstMoment |
| end |
| end |
|
|
| end OAI |
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