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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 | import OAI.NumberTheory.Jacobsthal.Estimates.LargeGapCountErrors
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
namespace NumberTheoryLean.CompactSmallCountErrors
open FinitePathGeometry PrimeHistories PrimeBinMembership ReferenceAdmission StrongSourceFamilies
open ActualCountErrorEdges ActualCountErrorMass SourcePrimeProductData
open LogarithmicBinPartition ErdosInverseCounts
open ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.PrimePrefixTail
theorem uniform_compact_small_count_errors (K d : ℝ) (hK : 3 ≤ K) (hd : 0 < d) :
∃ C B₀ w₀ : ℝ,0 < C ∧ 3 ≤ B₀ ∧ 1 < w₀ ∧ ∀ B w top : ℝ,B₀ ≤ B → w₀ ≤ w →
w < top → Real.log B ≤ d*Real.log w → ∀ z : Node,
z.side=.even → 199/100 ≤ z.ratio → z.ratio ≤ 23/10 → Consistent z → z.cutoff=B → z.closed=true → w^B=top →
∀ Y : ℕ,0 < Y → ∀ residue : ℕ → ℕ,∀ eps : ℝ,0 ≤ eps → ∀ F : Finset (List ℕ),
F ⊆ referencePrefixes w (sourcePrimeSet w top) z.side z.gap →
(∀ ps∈F,(terminal w z ps).gap ≤ K ∧
|wordCountError Y (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) residue (SmallSieveFinite.smallEuler ⌊w⌋₊) ps| ≤ eps) →
(B^2/((Y:ℝ)*SmallSieveFinite.smallEuler ⌊w⌋₊))*(∑ ps∈F,
|(modulusCount Y (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) residue ps.prod:ℝ)-
((Y:ℝ)/(ps.prod:ℝ))*SmallSieveFinite.smallEuler ⌊w⌋₊|) ≤ C*eps := by
obtain ⟨C,B₀,W,hC,hB₀,hW,hMass⟩ := uniform_source_family_mass K d hK hd
refine ⟨C,B₀,max W 2,hC,hB₀,hW.trans_le (le_max_left _ _),?_⟩
intro B w top hB hw₀ htop hcomp z hi h199 h23 hz hcut hclosed hpower Y hY residue eps heps F hF hGood
have hw : 1 < w := hW.trans_le ((le_max_left _ _).trans hw₀)
have hw2 : 2 ≤ w := (le_max_right _ _).trans hw₀
have hV : 0 < SmallSieveFinite.smallEuler ⌊w⌋₊ :=
(inv_pos.mpr (zero_lt_one.trans hw)).trans_le (SmallSieveFinite.smallEuler_floor_ge_inv w hw2)
have hp : ∀ ps∈F,0 < ps.prod := fun ps hps => (reference_word_product_data hw htop z.side z.gap ps (hF hps)).1
rw [finite_count_error_weight B Y _ residue _ hY hV F hp]
have he : (∑ ps∈F,prefixWeight ps*|wordCountError Y (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) residue
(SmallSieveFinite.smallEuler ⌊w⌋₊) ps|) ≤ (∑ ps∈F,prefixWeight ps)*eps := by
rw [Finset.sum_mul]
apply Finset.sum_le_sum
intro ps hps
exact mul_le_mul_of_nonneg_left (hGood ps hps).2 (prefixWeight_nonneg ps)
have hsum := hMass B w top hB ((le_max_left _ _).trans hw₀) htop hcomp z hi h199 h23 hz hcut hclosed hpower F hF
(fun ps hps => (hGood ps hps).1)
have hh := mul_le_mul_of_nonneg_left he (sq_nonneg B)
have hc := mul_le_mul_of_nonneg_right hsum heps
nlinarith
end NumberTheoryLean.CompactSmallCountErrors
end Erdos970
end OAI
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