| import OAI.NumberTheory.TwoPoint.Fourier.MinorArcTheorem |
| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTBandParameters |
| import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics |
|
|
| namespace OAI |
|
|
| /-! First-band parameters. A larger fixed lower-endpoint power absorbs |
| the explicit factorial constants; it changes only the absolute constant. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Filter |
|
|
| noncomputable def minorArcFirstLower (A : ℕ) (H : ℝ) : ℝ := (Real.log H) ^ (5 * A) |
|
|
| noncomputable def minorArcFirstUpper (H : ℝ) : ℝ := H / (Real.log H) ^ 15 |
|
|
| lemma minor_arc_first_lower_log (A : ℕ) (H : ℝ) : |
| Real.log (minorArcFirstLower A H) = 5 * (A : ℝ) * Real.log (Real.log H) := by |
| simp only [minorArcFirstLower, Real.log_pow, Nat.cast_mul, Nat.cast_ofNat] |
|
|
| lemma minor_arc_first_band_budget (A : ℕ) (hA : 500000 ≤ A) (H : ℝ) |
| (hloglog : 1 ≤ Real.log (Real.log H)) (hQ : 1 < minorArcFirstUpper H) |
| (hQH : minorArcFirstUpper H ≤ H) : |
| 8192 * (Real.log (Real.log (minorArcFirstUpper H)) + 1) ≤ |
| (1 / 100 : ℝ) * Real.log (minorArcFirstLower A H) := by |
| have hmono := Real.log_le_log (Real.log_pos hQ) |
| (Real.log_le_log (by linarith : 0 < minorArcFirstUpper H) hQH) |
| have hA' : (500000 : ℝ) ≤ A := by exact_mod_cast hA |
| have hm := mul_le_mul_of_nonneg_right hA' (by linarith : 0 ≤ Real.log (Real.log H)) |
| rw [minor_arc_first_lower_log] |
| nlinarith |
|
|
| lemma minor_arc_first_band_density (A : ℕ) (H : ℝ) |
| (hL : 0 < Real.log H) (hLL : 0 ≤ Real.log (Real.log H)) |
| (hQ : Real.log H / 2 ≤ Real.log (minorArcFirstUpper H)) : |
| Real.log (minorArcFirstLower A H) / Real.log (minorArcFirstUpper H) ≤ |
| 10 * (A : ℝ) * Real.log (Real.log H) / Real.log H := by |
| rw [minor_arc_first_lower_log] |
| calc |
| _ ≤ (5 * (A : ℝ) * Real.log (Real.log H)) / (Real.log H / 2) := |
| div_le_div_of_nonneg_left (by positivity) (by positivity) hQ |
| _ = _ := by ring |
|
|
| theorem minor_arc_first_band_scale (A R₀ : ℕ) (hA : 2 ≤ A) : |
| ∀ᶠ H : ℝ in atTop, 4 ≤ H ∧ 2 ≤ Real.log H ∧ 1 ≤ Real.log (Real.log H) ∧ |
| 4 ≤ minorArcFirstLower A H ∧ 2 * (R₀ : ℝ) ≤ minorArcFirstLower A H ∧ |
| 2 * (Real.log H) ^ 5 ≤ minorArcFirstLower A H ∧ |
| minorArcFirstLower A H ≤ minorArcFirstUpper H ∧ |
| minorArcFirstUpper H ≤ H / (Real.log H) ^ 5 ∧ |
| (Real.log H) / 2 ≤ Real.log (minorArcFirstUpper H) ∧ |
| 1 ≤ Real.log (minorArcFirstUpper H) := by |
| let E : ℕ := max (5 * A + 15) 30 |
| have hsmall := (Real.isLittleO_pow_log_id_atTop (n := E)).bound (by norm_num : (0 : ℝ) < 1) |
| have hlarge := Real.tendsto_log_atTop.eventually |
| (eventually_ge_atTop (max (4 : ℝ) (2 * R₀))) |
| have hloglog := (Real.tendsto_log_atTop.comp Real.tendsto_log_atTop).eventually |
| (eventually_ge_atTop (1 : ℝ)) |
| filter_upwards [hsmall, hlarge, hloglog, eventually_ge_atTop (4 : ℝ)] with H hs hl hll hH |
| have hL4 : 4 ≤ Real.log H := (le_max_left _ _).trans hl |
| have hLR : 2 * (R₀ : ℝ) ≤ Real.log H := (le_max_right _ _).trans hl |
| have hL1 : 1 ≤ Real.log H := by linarith |
| have hL0 : 0 < Real.log H := by linarith |
| have hH0 : 0 < H := by linarith |
| dsimp only [id] at hs |
| rw [Real.norm_eq_abs, abs_of_nonneg (pow_nonneg hL0.le E), Real.norm_eq_abs, |
| abs_of_pos hH0, one_mul] at hs |
| have hsA : Real.log H ^ (5 * A + 15) ≤ H := |
| (pow_le_pow_right₀ hL1 (le_max_left _ _)).trans hs |
| have hs30 : Real.log H ^ 30 ≤ H := |
| (pow_le_pow_right₀ hL1 (le_max_right _ _)).trans hs |
| have hP0 : 0 < minorArcFirstLower A H := by unfold minorArcFirstLower; positivity |
| have hPL : Real.log H ≤ minorArcFirstLower A H := |
| le_self_pow₀ hL1 (by omega : 5 * A ≠ 0) |
| have hPQ : minorArcFirstLower A H ≤ minorArcFirstUpper H := by |
| apply (le_div_iff₀ (pow_pos hL0 15)).mpr |
| simpa only [minorArcFirstLower, ← pow_add] using hsA |
| have hW2 : 2 ≤ Real.log H ^ 5 := |
| (by linarith : 2 ≤ Real.log H).trans (le_self_pow₀ hL1 (by norm_num : (5 : ℕ) ≠ 0)) |
| have hWP : 2 * Real.log H ^ 5 ≤ minorArcFirstLower A H := by |
| have hpow := pow_le_pow_right₀ (by linarith : 1 ≤ Real.log H ^ 5) hA |
| have hbase : 2 * Real.log H ^ 5 ≤ (Real.log H ^ 5) ^ 2 := by nlinarith |
| exact hbase.trans (by simpa only [pow_mul, minorArcFirstLower] using hpow) |
| have hQU : minorArcFirstUpper H ≤ H / Real.log H ^ 5 := |
| div_le_div_of_nonneg_left hH0.le (pow_pos hL0 5) |
| (pow_le_pow_right₀ hL1 (by norm_num : 5 ≤ 15)) |
| have h30 := Real.log_le_log (pow_pos hL0 30) hs30 |
| rw [Real.log_pow] at h30 |
| norm_num at h30 |
| have hlogQ : Real.log (minorArcFirstUpper H) = Real.log H - 15 * Real.log (Real.log H) := by |
| rw [minorArcFirstUpper, Real.log_div hH0.ne' (pow_ne_zero 15 hL0.ne'), Real.log_pow] |
| norm_num |
| have hhalf : Real.log H / 2 ≤ Real.log (minorArcFirstUpper H) := by rw [hlogQ]; linarith |
| exact ⟨hH, by linarith, hll, hL4.trans hPL, hLR.trans hPL, hWP, hPQ, hQU, |
| hhalf, by linarith⟩ |
|
|
| theorem minor_arc_first_band_density_eventually (A : ℕ) (hA : 2 ≤ A) : |
| ∀ᶠ H : ℝ in atTop, |
| Real.log (minorArcFirstLower A H) / Real.log (minorArcFirstUpper H) ≤ |
| 10 * (A : ℝ) * Real.log (Real.log H) / Real.log H := by |
| filter_upwards [minor_arc_first_band_scale A 0 hA] with H hH |
| obtain ⟨_, hL, hLL, _, _, _, _, _, hhalf, _⟩ := hH |
| exact minor_arc_first_band_density A H (by linarith) (by linarith) hhalf |
|
|
| theorem minor_arc_first_band_budget_eventually (A : ℕ) (hA : 500000 ≤ A) : |
| ∀ᶠ H : ℝ in atTop, |
| 8192 * (Real.log (Real.log (minorArcFirstUpper H)) + 1) ≤ |
| (1 / 100 : ℝ) * Real.log (minorArcFirstLower A H) := by |
| filter_upwards [minor_arc_first_band_scale A 0 (by omega)] with H hs |
| obtain ⟨hH, hL, hLL, hP, _hPR, _hPW, hPQ, hQU, _hhalf, _hQ⟩ := hs |
| apply minor_arc_first_band_budget A hA H hLL (by linarith) |
| exact hQU.trans (div_le_self (by linarith) (one_le_pow₀ (by linarith : 1 ≤ Real.log H))) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
