| import OAI.NumberTheory.Jacobsthal.Estimates.PatternCardGrowth |
| import OAI.NumberTheory.Jacobsthal.Estimates.SourceExtensionScale |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| open _root_.Filter |
| namespace ErdosVarianceLargeMap |
| open NumberTheoryLean ErdosVarianceSmallModel ErdosVarianceLargeCount ErdosLargeHeightBlocks |
| ErdosInversePrimeBin ErdosInverseBoxHeight ErdosInverseEuler |
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| theorem normalization_budgets (xi eps : ℝ) (hxi : 0 < xi) (heps : 0 < eps) : |
| ∀ᶠ R : ℝ in atTop,1 < R ∧ Real.log R ≤ R^((1 : ℝ)/40) ∧ |
| 16 ≤ eps*xi*R^((1 : ℝ)/20) := by |
| have hlog := (isLittleO_log_rpow_rpow_atTop (1 : ℝ) (by norm_num : (0 : ℝ) < 1/40)).bound zero_lt_one |
| filter_upwards [hlog,(tendsto_rpow_atTop (by norm_num : (0 : ℝ) < 1/20)).eventually_ge_atTop (16/(eps*xi)), |
| eventually_gt_atTop (1 : ℝ)] with R hlog hc hR |
| have hR0 : 0 < R := by linarith |
| refine ⟨hR,?_,?_⟩ |
| · simpa only [Real.rpow_one,Real.norm_eq_abs,abs_of_nonneg (Real.log_pos hR).le, |
| abs_of_nonneg (Real.rpow_nonneg hR0.le _),one_mul] using hlog |
| · have hh := (div_le_iff₀ (mul_pos heps hxi)).mp hc |
| nlinarith |
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| theorem source_pattern_error_normalized (alpha xi eps : ℝ) |
| (halpha : 0 < alpha) (hxi : 0 < xi) (hxi1 : xi ≤ 1) (heps : 0 < eps) : |
| ∀ᶠ z : ℝ in atTop,∀ (R theta : ℝ),z^alpha ≤ R → xi/4 ≤ theta → theta ≤ xi → |
| ∀ (H : ℕ) (C0 : ℤ),0 < H → |
| max (H : ℝ) (|(C0 : ℝ)|/R)/(H : ℝ) ≤ (sourceZ z)^12 → |
| ((blockCount C0 R theta H : ℝ)* |
| (patternCard (divisorModulus (sourceW z) H) (coprimeModulus (sourceW z) H) : ℝ)*R^((9 : ℝ)/10))/ |
| ((primeBin R theta).card : ℝ) ≤ eps := by |
| obtain ⟨R0,hR0⟩ := eventually_atTop.mp (normalization_budgets xi eps hxi heps) |
| filter_upwards [source_model_population alpha xi halpha hxi hxi1,source_patternCard_Z, |
| source_Z_power_R alpha (1/40) 14 halpha (by norm_num) (by decide), |
| sourceZ_tendsto_atTop.eventually_ge_atTop 2,(tendsto_rpow_atTop halpha).eventually_ge_atTop R0] |
| with z hpop hM hZpow hZ hzR |
| intro R theta hRlo hthetal hthetau H C0 hH hratio |
| have hp := hpop R theta hRlo hthetal hthetau |
| have hb := hR0 R (hzR.trans hRlo) |
| have hR : 0 < R := by linarith [hp.1] |
| have hlog : 0 < Real.log R := Real.log_pos hp.1 |
| have htheta : 0 ≤ theta := (by positivity : (0 : ℝ) ≤ xi/4).trans hthetal |
| let n : ℝ := blockCount C0 R theta H |
| let M : ℝ := patternCard (divisorModulus (sourceW z) H) (coprimeModulus (sourceW z) H) |
| have hn : n ≤ (sourceZ z)^13 := blockCount_source_bound C0 R theta (sourceZ z) H |
| hR htheta (hthetau.trans hxi1) hH hZ hratio |
| have hMZ : M ≤ sourceZ z := hM H |
| have hnM : n*M ≤ R^((1 : ℝ)/40) := by |
| calc |
| _ ≤ (sourceZ z)^13*sourceZ z := mul_le_mul hn hMZ (Nat.cast_nonneg _) (by positivity) |
| _ = (sourceZ z)^14 := by ring |
| _ ≤ _ := hZpow.2 R hRlo |
| have hprod : (n*M*R^((9 : ℝ)/10))*(16*Real.log R) ≤ 16*R^((19 : ℝ)/20) := by |
| have hpair := mul_le_mul hnM hb.2.1 hlog.le (Real.rpow_nonneg hR.le _) |
| have hm := mul_le_mul_of_nonneg_left hpair (show 0 ≤ 16*R^((9 : ℝ)/10) by positivity) |
| calc |
| _ ≤ 16*R^((9 : ℝ)/10)*(R^((1 : ℝ)/40)*R^((1 : ℝ)/40)) := by nlinarith |
| _ = 16*(R^((9 : ℝ)/10)*(R^((1 : ℝ)/40)*R^((1 : ℝ)/40))) := by ring |
| _ = _ := by rw [← Real.rpow_add hR,← Real.rpow_add hR];norm_num |
| have htarget : 16*R^((19 : ℝ)/20) ≤ eps*xi*R := by |
| have hh := mul_le_mul_of_nonneg_right hb.2.2 (Real.rpow_nonneg hR.le ((19 : ℝ)/20)) |
| calc |
| _ ≤ (eps*xi*R^((1 : ℝ)/20))*R^((19 : ℝ)/20) := hh |
| _ = _ := by rw [mul_assoc (eps*xi),← Real.rpow_add hR];norm_num |
| have herr : n*M*R^((9 : ℝ)/10) ≤ eps*(xi*R/(16*Real.log R)) := by |
| have hh := (le_div_iff₀ (show 0 < 16*Real.log R by positivity)).mpr (hprod.trans htarget) |
| have he : eps*(xi*R/(16*Real.log R)) = eps*xi*R/(16*Real.log R) := by ring |
| exact hh.trans_eq he.symm |
| have hlow : 0 < xi*R/(16*Real.log R) := by positivity |
| have hPC : 0 < ((primeBin R theta).card : ℝ) := hlow.trans_le hp.2.2 |
| apply (div_le_iff₀ hPC).mpr |
| exact herr.trans (mul_le_mul_of_nonneg_left hp.2.2 heps.le) |
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| end ErdosVarianceLargeMap |
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| end Erdos970 |
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| end OAI |
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