| import OAI.NumberTheory.TwoPoint.Bounds.ActualPaddingIntegerBounds |
| import OAI.NumberTheory.TwoPoint.Bounds.CoordinateProjection |
| import OAI.NumberTheory.TwoPoint.Bounds.NonnegativeProgressionAverage |
|
|
| namespace OAI |
|
|
| /-! Test the retained graph against bounded arithmetic functions. The |
| norm average uses the truncated integer padding estimate. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset Filter |
| open scoped Classical |
|
|
| lemma norm_inner_operator_le {V : Type*} [Fintype V] |
| (T : EuclideanSpace ℂ V →L[ℂ] EuclideanSpace ℂ V) (v : EuclideanSpace ℂ V) : |
| ‖inner ℂ v (T v)‖ ≤ ‖T‖ * ‖v‖ ^ 2 := by |
| calc |
| _ ≤ ‖v‖ * ‖T v‖ := norm_inner_le_norm _ _ |
| _ ≤ ‖v‖ * (‖T‖ * ‖v‖) := |
| mul_le_mul_of_nonneg_left (T.le_opNorm v) (norm_nonneg _) |
| _ = _ := by ring |
|
|
| theorem FiniteLaw.average_operator_test {Ω V : Type*} [Fintype Ω] [Fintype V] |
| (μ : FiniteLaw Ω) (T : Ω → EuclideanSpace ℂ V →L[ℂ] EuclideanSpace ℂ V) |
| (v : Ω → EuclideanSpace ℂ V) (B D Z δ : ℝ) (hB : 0 ≤ B) (hD : 0 ≤ D) |
| (hv : μ.average (fun x => ‖v x‖ ^ 2) ≤ Z) |
| (hbad : μ.probability (fun x => B < ‖T x‖) ≤ δ) |
| (hbound : ∀ x, B < ‖T x‖ → ‖inner ℂ (v x) (T x (v x))‖ ≤ D) : |
| μ.average (fun x => ‖inner ℂ (v x) (T x (v x))‖) ≤ B * Z + D * δ := by |
| have hp (x : Ω) : ‖inner ℂ (v x) (T x (v x))‖ ≤ |
| B * ‖v x‖ ^ 2 + D * (if B < ‖T x‖ then 1 else 0) := by |
| by_cases hx : B < ‖T x‖ |
| · rw [ite_eq_left hx, mul_one] |
| exact (hbound x hx).trans (le_add_of_nonneg_left (mul_nonneg hB (sq_nonneg _))) |
| · rw [ite_eq_right hx, mul_zero, add_zero] |
| exact (norm_inner_operator_le (T x) (v x)).trans |
| (mul_le_mul_of_nonneg_right (le_of_not_gt hx) (sq_nonneg _)) |
| calc |
| _ ≤ μ.average (fun x => B * ‖v x‖ ^ 2 + D * (if B < ‖T x‖ then 1 else 0)) := |
| μ.average_mono hp |
| _ = B * μ.average (fun x => ‖v x‖ ^ 2) + |
| D * μ.probability (fun x => B < ‖T x‖) := by |
| simp only [FiniteLaw.average, FiniteLaw.probability, mul_add, sum_add_distrib] |
| congr 1 <;> rw [mul_sum] <;> apply sum_congr rfl <;> intro x _ <;> ring |
| _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_left hv hB) |
| (mul_le_mul_of_nonneg_left hbad hD) |
|
|
| noncomputable def paddingTestVector {V : Type*} (Q : Finset ℕ) (L : ℝ) |
| (site : V → ℤ) (f : ℤ → ℂ) : EuclideanSpace ℂ V := |
| WithLp.toLp 2 (fun i => if actualPaddingDegreeCut Q L (site i) then |
| (actualPaddingVertex Q (site i) : ℂ) * f (site i) else 0) |
|
|
| lemma paddingTestVector_norm_sq_le {V : Type*} [Fintype V] |
| (Q : Finset ℕ) (L : ℝ) (site : V → ℤ) (f : ℤ → ℂ) (hf : ∀ n, ‖f n‖ ≤ 1) : |
| ‖paddingTestVector Q L site f‖ ^ 2 ≤ |
| ∑ i, (actualPaddingVertex Q (site i)) ^ 2 * |
| (if actualPaddingDegreeCut Q L (site i) then 1 else 0) := by |
| rw [EuclideanSpace.norm_sq_eq] |
| apply sum_le_sum |
| intro i _ |
| by_cases hi : actualPaddingDegreeCut Q L (site i) |
| · simp only [paddingTestVector, WithLp.ofLp_toLp, ite_eq_left hi, norm_mul, |
| Complex.norm_real, Real.norm_eq_abs, abs_of_pos (actualPaddingVertex_pos Q _), mul_one] |
| apply pow_le_pow_left₀ |
| (mul_nonneg (actualPaddingVertex_pos Q (site i)).le (norm_nonneg _)) _ 2 |
| simpa only [mul_one] using |
| mul_le_mul_of_nonneg_left (hf (site i)) (actualPaddingVertex_pos Q (site i)).le |
| · simp [paddingTestVector, hi] |
|
|
| theorem BravermanDepth22Input.eventually_padding_test_norm |
| (hBr : BravermanDepth22Input) : |
| ∃ A : ℕ, 1000 ≤ A ∧ ∀ᶠ L : ℝ in atTop, |
| ∀ (Q : Finset ℕ), (∀ p ∈ Q, p.Prime) → |
| (∀ p ∈ Q, (p : ℝ) ≤ Real.exp L) → |
| ∀ {V : Type*} [Fintype V] (site : V → ℤ) (f : ℤ → ℂ), (∀ n, ‖f n‖ ≤ 1) → |
| ∀ a N : ℕ, Real.exp (L ^ A / 2) ≤ (N : ℝ) → |
| uniformAverage (fun x : Fin N => ‖paddingTestVector Q L |
| (fun i => ((a + x.val : ℕ) : ℤ) + site i) f‖ ^ 2) ≤ |
| 2 * Fintype.card V * paddingTiltNormalizer Q := by |
| obtain ⟨A, hA, hb⟩ := hBr.eventually_actual_padding_weight_sum |
| refine ⟨A, hA, ?_⟩ |
| filter_upwards [hb] with L hb |
| intro Q hQ hupper V _ site f hf a N hN |
| classical |
| calc |
| _ ≤ uniformAverage (fun x : Fin N => ∑ i : V, |
| (actualPaddingVertex Q (((a + x.val : ℕ) : ℤ) + site i)) ^ 2 * |
| if actualPaddingDegreeCut Q L (((a + x.val : ℕ) : ℤ) + site i) then 1 else 0) := by |
| unfold uniformAverage |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _) |
| exact sum_le_sum (fun x _ => paddingTestVector_norm_sq_le Q L _ f hf) |
| _ = ∑ i : V, uniformAverage (fun x : Fin N => |
| (actualPaddingVertex Q (((a + x.val : ℕ) : ℤ) + site i)) ^ 2 * |
| if actualPaddingDegreeCut Q L (((a + x.val : ℕ) : ℤ) + site i) then 1 else 0) := by |
| unfold uniformAverage |
| rw [sum_comm, sum_div] |
| _ ≤ _ := by simpa only [card_univ] using hb Q hQ hupper univ site a N hN |
|
|
| theorem BravermanDepth22Input.eventually_padding_test_progression_norm |
| (hBr : BravermanDepth22Input) : |
| ∃ A : ℕ, 1000 ≤ A ∧ ∀ᶠ L : ℝ in atTop, |
| ∀ (Q : Finset ℕ), (∀ p ∈ Q, p.Prime) → |
| (∀ p ∈ Q, (p : ℝ) ≤ Real.exp L) → |
| ∀ {V : Type*} [Fintype V] (site : V → ℤ) (f : ℤ → ℂ), (∀ n, ‖f n‖ ≤ 1) → |
| ∀ a l N : ℕ, 0 < l → Real.exp (L ^ A / 2) ≤ (N : ℝ) → |
| uniformAverage (fun x : Fin N => ‖paddingTestVector Q L |
| (fun i => ((a + l * x.val : ℕ) : ℤ) + site i) f‖ ^ 2) ≤ |
| (l : ℝ) * (2 * Fintype.card V * paddingTiltNormalizer Q) := by |
| obtain ⟨A, hA, hb⟩ := hBr.eventually_padding_test_norm |
| refine ⟨A, hA, ?_⟩ |
| filter_upwards [hb] with L hb |
| intro Q hQ hupper V _ site f hf a l N hl hN |
| have hlr : (1 : ℝ) ≤ l := by exact_mod_cast hl |
| have hNl : Real.exp (L ^ A / 2) ≤ (l * N : ℕ) := by |
| rw [Nat.cast_mul] |
| exact hN.trans (by nlinarith [Nat.cast_nonneg (α := ℝ) N]) |
| have hp := uniformAverage_progression_le |
| (fun n => ‖paddingTestVector Q L (fun i => (n : ℤ) + site i) f‖ ^ 2) |
| (fun n => sq_nonneg _) a l N hl |
| exact hp.trans (mul_le_mul_of_nonneg_left |
| (hb Q hQ hupper site f hf a (l * N) hNl) (Nat.cast_nonneg _)) |
|
|
| lemma projected_paddingTestVector_norm_sq_le {V : Type*} [Fintype V] [DecidableEq V] |
| (Q : Finset ℕ) (L : ℝ) (site : V → ℤ) (f : ℤ → ℂ) |
| (keep : V → Prop) : |
| ‖coordinateProjection keep (paddingTestVector Q L site f)‖ ^ 2 ≤ |
| ‖paddingTestVector Q L site f‖ ^ 2 := |
| (sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).mpr (coordinateProjection_norm_le _ _) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
