File size: 4,272 Bytes
a72760f | 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.PiExponent.Analysis.AnalyticRowSeries
import OAI.NumberTheory.PiExponent.Analysis.CollisionExponential
import OAI.NumberTheory.PiExponent.Analysis.CollisionLimit
namespace OAI
open scoped BigOperators
namespace PiExponent.AnalyticCollision
theorem translated_row_determinant_bound
{ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ]
(group : ι → κ) (f : κ → ι → ℂ → ℂ)
(center : ι → ℂ) (ell : ι → ℕ)
(R : NNReal) (hR : 0 < R) (D s : ℝ) (hD : 0 ≤ D)
(hs0 : 0 ≤ s) (hs1 : s < 1) (hss : s * s = 1 / 2)
(hf : ∀ a j, Differentiable ℂ (f a j))
(hfD : ∀ a j z, z ∈ Metric.sphere (0 : ℂ) (R : ℝ) → ‖f a j z‖ ≤ D)
(hc : ∀ i, ‖center i‖ + 3 / 4 ≤ (R : ℝ) / 2) :
‖Matrix.det (fun i j => rowTest (ell i)
(fun t => f (group i) j (center i + Complex.log (1 + t))))‖ ≤
((Fintype.card ι).factorial * D ^ Fintype.card ι * (∏ i, (2 : ℝ) ^ ell i)) *
s ^ (∑ a, (Collision.multiplicity Finset.univ group a).choose 2) *
((1 - s)⁻¹) ^ Fintype.card ι := by
let B : ι → ℕ → ℂ := fun i d => rowTest (ell i)
(fun t => ((center i + Complex.log (1 + t)) / (R : ℂ)) ^ d)
let C : κ → ℕ → ι → ℂ := fun a d j => normalizedTaylorCoeff (f a j) R d
apply Collision.infinite_collision_bound group B C D s (fun i => (2 : ℝ) ^ ell i)
hD hs0 hs1 (fun _ => by positivity)
· intro a d j
exact norm_normalizedTaylorCoeff_le (by exact_mod_cast hR) (hfD a j) d
· intro i d
dsimp [B]
rw [hss]
exact norm_rowTest_shifted_power_le (by exact_mod_cast hR) (hc i) (ell i) d
· intro i j
exact hasSum_translated_row (hf (group i) j) hR hD (hfD (group i) j) (hc i) (ell i)
theorem translated_row_determinant_bound_sum_order
{ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ]
(group : ι → κ) (f : κ → ι → ℂ → ℂ)
(center : ι → ℂ) (ell : ι → ℕ)
(R : NNReal) (hR : 0 < R) (D s : ℝ) (hD : 0 ≤ D)
(hs0 : 0 ≤ s) (hs1 : s < 1) (hss : s * s = 1 / 2)
(hf : ∀ a j, Differentiable ℂ (f a j))
(hfD : ∀ a j z, z ∈ Metric.sphere (0 : ℂ) (R : ℝ) → ‖f a j z‖ ≤ D)
(hc : ∀ i, ‖center i‖ + 3 / 4 ≤ (R : ℝ) / 2) :
‖Matrix.det (fun i j => rowTest (ell i)
(fun t => f (group i) j (center i + Complex.log (1 + t))))‖ ≤
((Fintype.card ι).factorial * D ^ Fintype.card ι * (2 : ℝ) ^ (∑ i, ell i)) *
s ^ (∑ a, (Collision.multiplicity Finset.univ group a).choose 2) *
((1 - s)⁻¹) ^ Fintype.card ι := by
simpa only [Finset.prod_pow_eq_pow_sum] using
translated_row_determinant_bound group f center ell R hR D s hD hs0 hs1 hss hf hfD hc
theorem translated_row_determinant_exp_bound
{ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ]
(group : ι → κ) (f : κ → ι → ℂ → ℂ)
(center : ι → ℂ) (ell : ι → ℕ)
(R : NNReal) (hR : 0 < R) {H A v : ℝ} (hH : 0 < H)
(hf : ∀ a j, Differentiable ℂ (f a j))
(hfD : ∀ a j z, z ∈ Metric.sphere (0 : ℂ) (R : ℝ) →
‖f a j z‖ ≤ Real.exp (H * A))
(hc : ∀ i, ‖center i‖ + 3 / 4 ≤ (R : ℝ) / 2)
(hell : ((∑ i, ell i : ℕ) : ℝ) ≤ (Fintype.card ι : ℝ) * H / v) :
‖Matrix.det (fun i j => rowTest (ell i)
(fun t => f (group i) j (center i + Complex.log (1 + t))))‖ ≤
Real.exp (-(Real.log 2 / 4) *
(∑ a, (Collision.multiplicity Finset.univ group a : ℝ) ^ 2) +
(Fintype.card ι : ℝ) * H *
(A + Real.log 2 / v + Collision.collisionRemainder (Fintype.card ι) H)) := by
have hfinite := translated_row_determinant_bound_sum_order group f center ell R hR
(Real.exp (H * A)) Collision.collisionRatio (Real.exp_pos _).le
Collision.collisionRatio_pos.le Collision.collisionRatio_lt_one
Collision.collisionRatio_mul_self hf hfD hc
exact hfinite.trans (Collision.collision_factor_le_exp_normalized (Fintype.card ι)
(∑ a, (Collision.multiplicity Finset.univ group a).choose 2) (∑ i, ell i)
hH hell (Collision.twice_collision_count_eq group))
end PiExponent.AnalyticCollision
end OAI
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