| import OAI.NumberTheory.Jacobsthal.Estimates.LargeGapCountErrors |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| namespace NumberTheoryLean.CompactSmallCountErrors |
| open FinitePathGeometry PrimeHistories PrimeBinMembership ReferenceAdmission StrongSourceFamilies |
| open ActualCountErrorEdges ActualCountErrorMass SourcePrimeProductData |
| open LogarithmicBinPartition ErdosInverseCounts |
| open ErdosPrimeInputs.PrimePrefixMass ErdosPrimeInputs.PrimePrefixTail |
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| 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 |
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| end Erdos970 |
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| end OAI |
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