File size: 4,259 Bytes
f89a930 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.NumberTheory.Jacobsthal.Estimates.PatternCardGrowth
import OAI.NumberTheory.Jacobsthal.Estimates.SourceExtensionScale
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
open _root_.Filter
namespace ErdosVarianceLargeMap
open NumberTheoryLean ErdosVarianceSmallModel ErdosVarianceLargeCount ErdosLargeHeightBlocks
ErdosInversePrimeBin ErdosInverseBoxHeight ErdosInverseEuler
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
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)
end ErdosVarianceLargeMap
end Erdos970
end OAI
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