| 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)] |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
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| end OAI |
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|
