File size: 2,966 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 | 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
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