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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 80 81 82 83 | import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticNormRate
import OAI.NumberTheory.Ostmann.Characters.FourierWindowDecay
import OAI.NumberTheory.Ostmann.Construction.SpectatorBulkScale
namespace OAI
/-! # The frequency budget after return to the original prime law -/
namespace Ostmann
open Filter
open scoped SchwartzMap
theorem arithmetic_prime_page_spectator_cost (n m : ℕ) :
((2 : ℝ) ^ (2 ^ n * m) * 2 * 3 ^ (2 ^ n * m)) * 2 ^ (2 ^ n * m) =
2 * Real.exp (Real.log 12 * (2 ^ n : ℕ) * m) := by
have hbase : (2 : ℝ) * 3 * 2 = 12 := by norm_num
calc
_ = 2 * (((2 : ℝ) * 3 * 2) ^ (2 ^ n * m)) := by
simp only [mul_pow]
ring
_ = _ := by
rw [hbase]
congr 1
rw [show Real.log 12 * (2 ^ n : ℕ) * m =
((2 ^ n * m : ℕ) : ℝ) * Real.log 12 by push_cast; ring,
Real.exp_nat_mul, Real.exp_log (by norm_num : (0 : ℝ) < 12)]
/-- The prime-cell mixture adds one factor two per bulk coordinate. This
changes the constant in the exponential from log six to log twelve. -/
theorem arithmetic_prime_norm_budget_rate (n : ℕ) (A C ε : ℝ)
(hA : 0 ≤ A) (hC : 0 ≤ C) (hε : 0 < ε) :
∀ᶠ m : ℝ in atTop, ∃ D : ℝ, 0 ≤ D ∧
(∀ q : ℕ, q ≠ 0 → (q : ℝ) ≤ Real.exp (2 * C * m) → (q.divisors.card : ℝ) ≤ D) ∧
∀ N V : ℕ, ∀ Δ : ℝ, (N : ℝ) ≤ Real.exp (C * m) →
(V : ℝ) ≤ Real.exp (Δ + Real.sqrt (4 * m)) →
(A * Real.exp (-(2 ^ n : ℕ) * Δ)) *
((2 * Real.exp (Real.log 12 * (2 ^ n : ℕ) * m)) *
((8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) *
(2 * (V : ℝ)) ^ (2 * 2 ^ n))) ≤
Real.exp ((2 ^ n : ℕ) * Δ + Real.log 12 * (2 ^ n : ℕ) * m + ε * m) := by
have hlog : Real.log 12 = Real.log 2 + Real.log 6 := by
rw [← Real.log_mul (by norm_num : (2 : ℝ) ≠ 0) (by norm_num : (6 : ℝ) ≠ 0)]
norm_num
filter_upwards [arithmetic_norm_budget_rate n A C ε hA hC hε] with m hm
obtain ⟨D, hD, hdiv, hcost⟩ := hm
refine ⟨D, hD, hdiv, ?_⟩
intro N V Δ hN hV
have h := mul_le_mul_of_nonneg_left (hcost N V Δ hN hV)
(Real.exp_pos (Real.log 2 * (2 ^ n : ℕ) * m)).le
convert h using 1 <;> rw [hlog] <;>
simp only [add_mul, Real.exp_add] <;> ring
/-- Specialization to the actual even bulk size, with the literal window and
Page/spectator constants of the prime norm estimate. -/
theorem arithmetic_prime_fourier_budget_rate (ψ : 𝓢(ℝ, ℂ)) (n k : ℕ) (hk : 0 < k)
(B C ε : ℝ) (hC : 0 ≤ C) (hε : 0 < ε) :
∀ᶠ L : ℝ in atTop, let m := spectatorBulkCount k L
∃ D : ℝ, 0 ≤ D ∧
(∀ q : ℕ, q ≠ 0 → (q : ℝ) ≤ Real.exp (2 * C * m) → (q.divisors.card : ℝ) ≤ D) ∧
∀ N V : ℕ, ∀ Δ : ℝ, (N : ℝ) ≤ Real.exp (C * m) →
(V : ℝ) ≤ Real.exp (Δ + Real.sqrt (4 * m)) →
(((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) *
B ^ (2 ^ n - 1)) ^ 2 *
(((2 : ℝ) ^ (2 ^ n * m) * 2 * 3 ^ (2 ^ n * m)) *
((8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) *
(2 * (V : ℝ)) ^ (2 * 2 ^ n)))) * 2 ^ (2 ^ n * m) ≤
Real.exp ((2 ^ n : ℕ) * Δ + Real.log 12 * (2 ^ n : ℕ) * m + ε * m) := by
let A : ℝ := ((SchwartzMap.seminorm ℝ 0 0 ψ) ^ (2 ^ n) * B ^ (2 ^ n - 1)) ^ 2
have hrate := (spectatorBulkCount_tendsto k hk).eventually
(arithmetic_prime_norm_budget_rate n A C ε (sq_nonneg _) hC hε)
filter_upwards [hrate] with L hrate
obtain ⟨D, hD, hdiv, hcost⟩ := hrate
refine ⟨D, hD, hdiv, ?_⟩
intro N V Δ hN hV
have h := hcost N V Δ hN hV
rw [← arithmetic_prime_page_spectator_cost n (spectatorBulkCount k L)] at h
rw [fourier_window_pair_exp]
exact le_trans (le_of_eq (by dsimp only [A]; ring)) h
end Ostmann
end OAI
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