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f4b3ad1 | 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 | import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticPatternErrorRate
import OAI.NumberTheory.Ostmann.Arithmetic.PatternFrequencyNormSum
namespace OAI
/-! # The diagonal budget after both giant and bulk comparisons -/
namespace Ostmann
open Filter
open scoped Classical BigOperators
/-- The actual pattern sum and paired frequency sum have an absolute bulk
constant. The cutoff includes the constructed bottom reserve `2 sqrt(m)`.
The five comparison errors retain their saving after the full history count. -/
theorem arithmetic_diagonal_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)) +
4 * 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) +
5 * 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_five_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
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