| import OAI.NumberTheory.TwoPoint.Fourier.MajorArcDivisorBounds |
| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTArcRate |
|
|
| namespace OAI |
|
|
| /-! The common major/minor-arc parameter and its quantitative rates. |
| The fifth-root working length retains the published logarithmic error. -/ |
| namespace TwoPointCorrelations |
|
|
| open Filter |
|
|
| noncomputable def majorArcParameter (L H M : ℝ) : ℝ := |
| min ((Real.log H)^5) (min (L^(1/125:ℝ)) (Real.exp (M/3))) |
|
|
| lemma major_arc_parameter_bounds {L H M : ℝ} (hL : 1 ≤ L) |
| (hH : 1 ≤ Real.log H) (hM : 0 ≤ M) : |
| 1 ≤ majorArcParameter L H M ∧ |
| majorArcParameter L H M ≤ (Real.log H)^5 ∧ |
| majorArcParameter L H M ≤ L^(1/125:ℝ) ∧ |
| majorArcParameter L H M ≤ Real.exp (M/3) := by |
| exact ⟨le_min (one_le_pow₀ hH) (le_min |
| (Real.one_le_rpow hL (by norm_num)) (Real.one_le_exp (by positivity))), |
| min_le_left _ _, (min_le_right _ _).trans (min_le_left _ _), |
| (min_le_right _ _).trans (min_le_right _ _)⟩ |
|
|
| noncomputable def majorArcWorkingError (W : ℝ) : ℝ := |
| (1+Real.log W)*W^(-1/5:ℝ) |
|
|
| lemma major_arc_working_short {H : ℝ} (hH : 1 ≤ Real.log H) |
| (hHH : 1 ≤ Real.log (Real.log H)) : |
| majorArcWorkingError ((Real.log H)^5) ≤ 6*Real.log (Real.log H)/Real.log H := by |
| have hL : 0 < Real.log H := by linarith |
| have hp : ((Real.log H)^5)^(-1/5:ℝ) = (Real.log H)⁻¹ := by |
| rw [← Real.rpow_natCast,← Real.rpow_mul hL.le] |
| norm_num [Real.rpow_neg_one] |
| unfold majorArcWorkingError |
| rw [hp,Real.log_pow] |
| rw [mul_inv_le_iff₀ hL] |
| field_simp |
| norm_num |
| linarith |
|
|
| lemma major_arc_working_exponential {M : ℝ} (hM : 0 ≤ M) : |
| majorArcWorkingError (Real.exp (M/3)) ≤ 21*Real.exp (-M/20) := by |
| have hh := Real.add_one_le_exp (M/60) |
| have hcoef : 1+M/3 ≤ 21*Real.exp (M/60) := by |
| have he : 1 ≤ Real.exp (M/60) := Real.one_le_exp (by positivity) |
| linarith |
| unfold majorArcWorkingError |
| rw [Real.log_exp,Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp] |
| calc |
| _ ≤ (21*Real.exp (M/60))*Real.exp (M/3*(-1/5)) := |
| mul_le_mul_of_nonneg_right hcoef (Real.exp_pos _).le |
| _ = _ := by |
| rw [mul_assoc,← Real.exp_add] |
| congr 2 |
| ring |
|
|
| lemma major_arc_working_outer : |
| ∀ᶠ L : ℝ in atTop, |
| majorArcWorkingError (L^(1/125:ℝ)) ≤ L^(-1/700:ℝ) := by |
| have hh := (isLittleO_log_rpow_atTop (show (0:ℝ)<3/17500 by norm_num)).bound |
| (show (0:ℝ)<1/2 by norm_num) |
| have hg := (tendsto_rpow_atTop (show (0:ℝ)<3/17500 by norm_num)).eventually |
| (eventually_ge_atTop (2:ℝ)) |
| filter_upwards [hh,hg,eventually_ge_atTop (1:ℝ)] with L hh hg hL |
| have hL0 : 0 < L := by linarith |
| rw [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hL),Real.norm_eq_abs, |
| abs_of_nonneg (Real.rpow_nonneg hL0.le _)] at hh |
| have hp : (L^(1/125:ℝ))^(-1/5:ℝ)=L^(-1/625:ℝ) := by |
| rw [← Real.rpow_mul hL0.le] |
| norm_num |
| unfold majorArcWorkingError |
| rw [Real.log_rpow hL0,hp] |
| calc |
| _ ≤ L^(3/17500:ℝ)*L^(-1/625:ℝ) := by |
| apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg hL0.le _) |
| nlinarith [Real.log_nonneg hL] |
| _ = _ := by rw [← Real.rpow_add hL0]; norm_num |
|
|
| lemma major_arc_working_min {a b : ℝ} (ha : 1 ≤ a) (hb : 1 ≤ b) : |
| majorArcWorkingError (min a b) ≤ majorArcWorkingError a+majorArcWorkingError b := by |
| have hfa : 0 ≤ majorArcWorkingError a := |
| mul_nonneg (by linarith [Real.log_nonneg ha]) (Real.rpow_nonneg (by linarith) _) |
| have hfb : 0 ≤ majorArcWorkingError b := |
| mul_nonneg (by linarith [Real.log_nonneg hb]) (Real.rpow_nonneg (by linarith) _) |
| rcases le_total a b with hab | hba |
| · rw [min_eq_left hab] |
| exact le_add_of_nonneg_right hfb |
| · rw [min_eq_right hba] |
| exact le_add_of_nonneg_left hfa |
|
|
| lemma major_arc_parameter_rate : |
| ∀ᶠ L : ℝ in atTop, ∀ H M : ℝ, 1 ≤ Real.log H → |
| 1 ≤ Real.log (Real.log H) → 0 ≤ M → |
| majorArcWorkingError (majorArcParameter L H M) ≤ |
| 6*Real.log (Real.log H)/Real.log H + L^(-1/700:ℝ)+21*Real.exp (-M/20) := by |
| filter_upwards [major_arc_working_outer,eventually_ge_atTop (1:ℝ)] with L houter hL |
| intro H M hH hHH hM |
| have ha : 1 ≤ (Real.log H)^5 := one_le_pow₀ hH |
| have hb : 1 ≤ L^(1/125:ℝ) := Real.one_le_rpow hL (by norm_num) |
| have hc : 1 ≤ Real.exp (M/3) := Real.one_le_exp (by positivity) |
| have hi := major_arc_working_min hb hc |
| have hj := major_arc_working_min ha (le_min hb hc) |
| change majorArcWorkingError (majorArcParameter L H M) ≤ _ at hj |
| linarith [major_arc_working_short hH hHH,major_arc_working_exponential hM] |
|
|
| theorem major_arc_sharp_energy_rate : |
| ∀ᶠ L : ℝ in atTop, ∀ W : ℝ, 1 ≤ W → W ≤ L^(1/125:ℝ) → |
| Real.sqrt W*(1+Real.log W)*Real.sqrt (Real.log L/L^(1/80:ℝ)) ≤ |
| 101*L^(-1/700:ℝ) := by |
| have hs := (isLittleO_log_rpow_atTop (show (0:ℝ)<1/1000 by norm_num)).bound |
| (show (0:ℝ)<1 by norm_num) |
| filter_upwards [hs,eventually_ge_atTop (1:ℝ)] with L hs hL |
| have hL0 : 0 < L := by linarith |
| rw [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg hL),Real.norm_eq_abs, |
| abs_of_nonneg (Real.rpow_nonneg hL0.le _),one_mul] at hs |
| intro W hW hWL |
| have hW0 : 0 < W := by linarith |
| have hmod : Real.sqrt W*W^(1/100:ℝ) ≤ L^(51/12500:ℝ) := by |
| rw [mrt_sqrt_modulus_power hW0] |
| calc |
| _ ≤ (L^(1/125:ℝ))^(51/100:ℝ) := Real.rpow_le_rpow hW0.le hWL (by norm_num) |
| _ = _ := by rw [← Real.rpow_mul hL0.le]; norm_num |
| have hratio : Real.log L/L^(1/80:ℝ) ≤ L^(-23/2000:ℝ) := by |
| calc |
| _ ≤ L^(1/1000:ℝ)/L^(1/80:ℝ) := div_le_div_of_nonneg_right hs (by positivity) |
| _ = _ := by rw [← Real.rpow_sub hL0]; norm_num |
| have hroot : Real.sqrt (Real.log L/L^(1/80:ℝ)) ≤ L^(-23/4000:ℝ) := by |
| calc |
| _ ≤ Real.sqrt (L^(-23/2000:ℝ)) := Real.sqrt_le_sqrt hratio |
| _ = _ := by rw [Real.sqrt_eq_rpow,← Real.rpow_mul hL0.le]; norm_num |
| calc |
| _ ≤ Real.sqrt W*(101*W^(1/100:ℝ))*Real.sqrt (Real.log L/L^(1/80:ℝ)) := by |
| gcongr |
| exact major_arc_log_overhead hW |
| _ = 101*((Real.sqrt W*W^(1/100:ℝ))*Real.sqrt (Real.log L/L^(1/80:ℝ))) := by ring |
| _ ≤ 101*(L^(51/12500:ℝ)*L^(-23/4000:ℝ)) := by |
| gcongr |
| _ = 101*L^(-167/100000:ℝ) := by rw [← Real.rpow_add hL0]; norm_num |
| _ ≤ _ := mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hL (by norm_num)) (by norm_num) |
|
|
| lemma major_arc_sharp_distance_rate {W M : ℝ} (hW : 1 ≤ W) (hM : 0 ≤ M) |
| (hWM : W ≤ Real.exp (M/3)) : |
| Real.sqrt W*(1+Real.log W)*Real.exp (-2*M/5) ≤ 101*Real.exp (-M/20) := by |
| have hW0 : 0 < W := by linarith |
| calc |
| _ ≤ Real.sqrt W*(101*W^(1/100:ℝ))*Real.exp (-2*M/5) := by |
| gcongr |
| exact major_arc_log_overhead hW |
| _ = 101*(W^(51/100:ℝ)*Real.exp (-2*M/5)) := by |
| rw [← mrt_sqrt_modulus_power hW0] |
| ring |
| _ ≤ 101*((Real.exp (M/3))^(51/100:ℝ)*Real.exp (-2*M/5)) := by |
| gcongr |
| _ = 101*Real.exp (-23*M/100) := by |
| rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp,← Real.exp_add] |
| congr 2 |
| ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (Real.exp_le_exp.mpr (by nlinarith only [hM])) (by norm_num) |
|
|
|
|
| lemma major_arc_inverse_working_rate {W : ℝ} (hW : 1 ≤ W) : |
| Real.sqrt W*(1+Real.log W)*W⁻¹ ≤ majorArcWorkingError W := by |
| have hW0 : 0 < W := by linarith |
| calc |
| _ = (1+Real.log W)*W^(-1/2:ℝ) := by |
| rw [Real.sqrt_eq_rpow,← Real.rpow_neg_one W] |
| calc |
| _ = (1+Real.log W)*(W^(1/2:ℝ)*W^(-1:ℝ)) := by ring |
| _ = _ := by rw [← Real.rpow_add hW0]; norm_num |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (Real.rpow_le_rpow_of_exponent_le hW (by norm_num)) |
| (by linarith [Real.log_nonneg hW]) |
|
|
| theorem major_arc_weighted_energy_rate : |
| ∀ᶠ L : ℝ in atTop, ∀ W M : ℝ, 1 ≤ W → W ≤ L^(1/125:ℝ) → |
| 0 ≤ M → W ≤ Real.exp (M/3) → |
| Real.sqrt W*(1+Real.log W)* |
| (Real.exp (-2*M/5)+Real.sqrt (Real.log L/L^(1/80:ℝ))+W⁻¹) ≤ |
| 101*Real.exp (-M/20)+101*L^(-1/700:ℝ)+majorArcWorkingError W := by |
| filter_upwards [major_arc_sharp_energy_rate] with L henergy |
| intro W M hW hWL hM hWM |
| have hm := major_arc_sharp_distance_rate hW hM hWM |
| have he := henergy W hW hWL |
| have hi := major_arc_inverse_working_rate hW |
| calc |
| _ = Real.sqrt W*(1+Real.log W)*Real.exp (-2*M/5)+ |
| Real.sqrt W*(1+Real.log W)*Real.sqrt (Real.log L/L^(1/80:ℝ))+ |
| Real.sqrt W*(1+Real.log W)*W⁻¹ := by ring |
| _ ≤ _ := add_le_add (add_le_add hm he) hi |
|
|
| end TwoPointCorrelations |
|
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| end OAI |
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