| import OAI.NumberTheory.PiExponent.Analysis.AnalyticDeterminantCollision |
| import OAI.NumberTheory.PiExponent.Analysis.PeriodAnalytic |
| import OAI.NumberTheory.PiExponent.Polynomials.SimplexLog |
|
|
| namespace OAI |
|
|
| open scoped BigOperators Topology |
| open Filter |
|
|
| namespace PiExponent |
|
|
| noncomputable def holomorphicError (K : ℕ) (w0 v0 wstar : ℝ) : ℝ := |
| 100 * K / w0 + Real.log 2 / v0 + Real.log (200 * K) / wstar |
|
|
| theorem formal_period_collision_bound |
| {m : ℕ} {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] |
| (group : ι → κ) (transverse : κ → Fin m →₀ ℕ) |
| (j ell h : ι → ℕ) (a : ι → Fin m → ℕ) (w : Fin m → ℝ) |
| {K : ℕ} {H w0 v0 wstar : ℝ} |
| (hK : 0 < K) (hH : 0 < H) (hw0 : 0 < w0) (hws : 0 < wstar) |
| (hw : ∀ i, wstar ≤ w i) (hj : ∀ r, j r < K) |
| (hcol : ∀ c, w0 * h c + ∑ i, w i * a c i ≤ H) |
| (hell : ((∑ r, ell r : ℕ) : ℝ) ≤ (Fintype.card ι : ℝ) * H / v0) : |
| ‖Matrix.det (fun r c => PowerSeries.coeff (ell r) |
| ((MatrixTranslation.periodMonomial (j r) (2 * (Real.pi : ℂ) * Complex.I) |
| (h c) (a c)).coeff (transverse (group r))))‖ ≤ |
| Real.exp (-(Real.log 2 / 4) * |
| (∑ A, (Collision.multiplicity Finset.univ group A : ℝ) ^ 2) + |
| (Fintype.card ι : ℝ) * H * |
| (holomorphicError K w0 v0 wstar + Collision.collisionRemainder (Fintype.card ι) H)) := by |
| let R : NNReal := ⟨100 * (K : ℝ), by positivity⟩ |
| have hR : 0 < R := by change (0 : ℝ) < 100 * K; positivity |
| have hR1 : (1 : ℝ) ≤ R := by |
| have hK1 : (1 : ℝ) ≤ K := by exact_mod_cast hK |
| change (1 : ℝ) ≤ 100 * K |
| linarith |
| have hb := AnalyticCollision.translated_row_determinant_exp_bound |
| group (fun A c => PeriodAnalytic.columnFunction (h c) (a c) (transverse A)) |
| (fun r => (j r : ℂ) * (2 * (Real.pi : ℂ) * Complex.I)) ell R hR hH |
| (fun A c => PeriodAnalytic.differentiable_columnFunction _ _ _) |
| (A := (R : ℝ) / w0 + Real.log (2 * (R : ℝ)) / wstar) (v := v0) |
| (fun A c z hz => PeriodAnalytic.norm_columnFunction_le (h c) (a c) (transverse A) w |
| hR1 hH.le hw0 hws hw (hcol c) |
| (by simpa only [Metric.mem_sphere, dist_zero_right] using hz.le)) |
| (fun r => AnalyticCollision.period_center_radius_bound hK (hj r)) hell |
| have hRv : (R : ℝ) = 100 * K := rfl |
| rw [hRv] at hb |
| have he : (2 : ℝ) * (100 * K) = 200 * K := by ring |
| rw [he] at hb |
| simp_rw [PeriodAnalytic.periodMonomial_coeff_eq_rowTest] |
| convert! hb using 1 |
| congr 1 |
| unfold holomorphicError |
| ring |
|
|
| theorem tendsto_collisionRemainder_of_normalized_pow {M : ℝ → ℕ} {d : ℕ} {L : ℝ} |
| (hM : Tendsto (fun H : ℝ => (M H : ℝ) / H ^ d) atTop (𝓝 L)) (hL : 0 < L) : |
| Tendsto (fun H => Collision.collisionRemainder (M H) H) atTop (𝓝 0) := by |
| convert tendsto_log_add_div_of_normalized_pow hM hL |
| (Real.log 2 / 4 - Real.log (1 - Collision.collisionRatio)) using 1 |
| funext H |
| unfold Collision.collisionRemainder |
| congr 1 |
| ring |
|
|
| end PiExponent |
|
|
| end OAI |
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|
