| import Mathlib |
| import OAI.NumberTheory.PiExponent.Analysis.FiniteDeterminantCollision |
|
|
| namespace OAI |
|
|
| open scoped BigOperators Topology |
| open Filter |
|
|
| namespace PiExponent.Collision |
|
|
| theorem norm_det_le_of_entrywise_limit {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (A : ℕ → Matrix ι ι ℂ) (T : Matrix ι ι ℂ) {B : ℝ} |
| (hlim : ∀ i j, Tendsto (fun N => A N i j) atTop (𝓝 (T i j))) |
| (hbound : ∀ N, ‖(A N).det‖ ≤ B) : ‖T.det‖ ≤ B := by |
| have hmatrix : Tendsto A atTop (𝓝 T) := by |
| apply tendsto_pi_nhds.mpr |
| intro i |
| exact tendsto_pi_nhds.mpr (hlim i) |
| have hdet : Tendsto (fun N => (A N).det) atTop (𝓝 T.det) := |
| (continuous_id.matrix_det.tendsto T).comp hmatrix |
| exact le_of_tendsto' hdet.norm hbound |
|
|
| theorem norm_det_le_of_entrywise_hasSum {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (C : ι → ℕ → ι → ℂ) (T : Matrix ι ι ℂ) {B : ℝ} |
| (hsum : ∀ i j, HasSum (fun d => C i d j) (T i j)) |
| (hbound : ∀ N, ‖Matrix.det (fun i j => ∑ d ∈ Finset.range N, C i d j)‖ ≤ B) : |
| ‖T.det‖ ≤ B := by |
| exact norm_det_le_of_entrywise_limit |
| (fun N i j => ∑ d ∈ Finset.range N, C i d j) T |
| (fun i j => (hsum i j).tendsto_sum_nat) hbound |
|
|
| theorem norm_det_le_of_tested_series_hasSum |
| {ι κ : Type*} [Fintype ι] [DecidableEq ι] |
| (group : ι → κ) (coeff : κ → ℕ → ι → ℂ) (test : ι → ℕ → ℂ) |
| (T : Matrix ι ι ℂ) {B : ℝ} |
| (hsum : ∀ i j, HasSum (fun d => test i d * coeff (group i) d j) (T i j)) |
| (hbound : ∀ N, |
| ‖Matrix.det (fun i j => ∑ d ∈ Finset.range N, test i d * coeff (group i) d j)‖ ≤ B) : |
| ‖T.det‖ ≤ B := by |
| exact norm_det_le_of_entrywise_hasSum |
| (fun i d j => test i d * coeff (group i) d j) T hsum hbound |
|
|
| theorem infinite_collision_bound |
| {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] |
| (group : ι → κ) (B : ι → ℕ → ℂ) (C : κ → ℕ → ι → ℂ) |
| (D s : ℝ) (L : ι → ℝ) (hD : 0 ≤ D) (hs0 : 0 ≤ s) (hs1 : s < 1) |
| (hL : ∀ i, 0 ≤ L i) |
| (hC : ∀ a d j, ‖C a d j‖ ≤ D) |
| (hB : ∀ i d, ‖B i d‖ ≤ L i * (s * s) ^ d) |
| (T : Matrix ι ι ℂ) |
| (hsum : ∀ i j, HasSum (fun d => B i d * C (group i) d j) (T i j)) : |
| ‖T.det‖ ≤ |
| ((Fintype.card ι).factorial * D ^ Fintype.card ι * (∏ i, L i)) * |
| s ^ (∑ a, (multiplicity Finset.univ group a).choose 2) * |
| ((1 - s)⁻¹) ^ Fintype.card ι := by |
| apply norm_det_le_of_tested_series_hasSum group C B T hsum |
| intro N |
| have hmatrix : (fun i j => ∑ d ∈ Finset.range N, B i d * C (group i) d j) = |
| (fun i j => ∑ d : Fin N, B i d * C (group i) d j) := by |
| funext i j |
| exact (Fin.sum_univ_eq_sum_range (fun d : ℕ => B i d * C (group i) d j) N).symm |
| rw [hmatrix] |
| exact FiniteDeterminantCollision.finite_collision_bound group B C D s L hD hs0 hs1 |
| hL hC hB N |
|
|
| end PiExponent.Collision |
|
|
| end OAI |
|
|
