| import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticPatternErrorRate |
| import OAI.NumberTheory.Ostmann.Arithmetic.PatternFrequencyNormSum |
|
|
| namespace OAI |
|
|
| /-! # The full numerical norm bound with the original equality patterns -/ |
|
|
| namespace Ostmann |
| open Filter |
| open scoped Classical BigOperators |
|
|
| / |
| constant. The cutoff includes the constructed bottom reserve `2 sqrt(m)`. |
| Only the genuine comparison errors use the full number of histories. -/ |
| theorem arithmetic_pattern_norm_rate (ψ : SchwartzMap ℝ ℂ) (n k : ℕ) |
| (hk : 0 < k) (hn : n ≤ k) (B Cfreq Cprior ε : ℝ) |
| (hCfreq : 0 ≤ Cfreq) (hCprior : 1 ≤ Cprior) (hε : 0 < ε) |
| (hdepth : 8 * Cprior ≤ (k : ℝ) ^ 3) : |
| ∀ᶠ L : ℝ in atTop, let m := spectatorBulkCount k L |
| ∃ D : ℝ, 0 ≤ D ∧ |
| (∀ q : ℕ, q ≠ 0 → (q : ℝ) ≤ Real.exp (2 * Cfreq * m) → |
| (q.divisors.card : ℝ) ≤ D) ∧ |
| ∀ (S : Finset ℤ) (N V : ℕ) (Δ err : ℝ), |
| (S.card : ℝ) ≤ Real.exp (Cfreq * m) → |
| (∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N) → |
| (N : ℝ) ≤ Real.exp (Cfreq * m) → |
| (V : ℝ) ≤ Real.exp (Δ + Real.sqrt (4 * m)) → |
| 0 ≤ err → err ≤ Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) + |
| 3 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L)) → |
| let _ := sampleSetoidFintype (Bool × MovingSampleIndex n) |
| let K := ((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) * |
| B ^ (2 ^ n - 1)) ^ 2 * |
| ((2 : ℝ) ^ (2 ^ n * m) * 4 * 3 ^ (2 ^ n * m)) * 2 ^ (2 ^ n * m) |
| ∀ R : Setoid (Bool × MovingSampleIndex n) → FrequencyTree (S × S) n → ℂ, |
| (∀ s t, ‖R s t‖ ≤ |
| ((2 : ℝ) ^ Fintype.card (Quotient s) * |
| (Real.exp (Cprior * L)) ^ (4 * n * 2 ^ n - Fintype.card (Quotient s))) * |
| (err + K * (frequencyLeafWeight (pairedFrequencyLeaf S V) n t * |
| ((frequencySplitList S n t).map (pairFrequencySupportBound D)).prod))) → |
| ‖∑ s, ∑ t, R s t‖ ≤ |
| Real.exp ((2 ^ n : ℕ) * Δ + (Real.log 12 + 1) * (2 ^ n : ℕ) * m + ε * m) + |
| 4 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := by |
| filter_upwards [arithmetic_prime_pattern_budget_rate ψ n k hk hn B Cfreq Cprior ε |
| hCfreq hCprior hε hdepth, |
| movingPattern_four_errors_rate n k hk Cprior Cfreq hCprior hCfreq] |
| with L hbudget herrors |
| dsimp only |
| obtain ⟨D, hD, hdiv, hbudget⟩ := hbudget |
| refine ⟨D, hD, hdiv, ?_⟩ |
| intro S N V Δ err hcard hS hN hV herr herrbound R hR |
| have hdivN : ∀ q : ℕ, q ≠ 0 → q ≤ N ^ 2 → (q.divisors.card : ℝ) ≤ D := by |
| intro q hq hqN |
| apply hdiv q hq |
| calc |
| (q : ℝ) ≤ (N : ℝ) ^ 2 := by exact_mod_cast hqN |
| _ ≤ (Real.exp (Cfreq * spectatorBulkCount k L)) ^ 2 := |
| pow_le_pow_left₀ (Nat.cast_nonneg N) hN 2 |
| _ = Real.exp (2 * Cfreq * spectatorBulkCount k L) := by |
| rw [← Real.exp_nat_mul] |
| congr 1 |
| ring |
| have hsum := pattern_frequency_norm_sum_le S n N V D (Real.exp (Cprior * L)) |
| (((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) * |
| B ^ (2 ^ n - 1)) ^ 2 * |
| ((2 : ℝ) ^ (2 ^ n * spectatorBulkCount k L) * 4 * 3 ^ (2 ^ n * spectatorBulkCount k L)) * |
| 2 ^ (2 ^ n * spectatorBulkCount k L)) err hD (Real.exp_nonneg _) (by positivity) |
| herr hS hdivN R hR |
| have hcost := hbudget N V Δ hN hV |
| have he := (mul_le_mul_of_nonneg_left herrbound |
| (show 0 ≤ ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * |
| (max 2 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) * |
| (Fintype.card (FrequencyTree (S × S) n) : ℝ) by positivity)).trans (herrors S hcard) |
| apply hsum.trans |
| nlinarith only [hcost, he] |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
