| import OAI.NumberTheory.TwoPoint.Walks.BlockEdgeForm |
| import OAI.NumberTheory.TwoPoint.Bounds.FiniteAverages |
|
|
| namespace OAI |
|
|
| /-! Normalize an averaged quadratic form and transfer its bound to the |
| retained prefix, with explicit displacement and boundary errors. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
|
|
| lemma edgeBlockSum_norm_average_le {E : Type*} [Fintype E] |
| (r : E → ℕ) (c : E → ℕ → ℂ) (M N : ℕ) : |
| ‖((∑ t ∈ range N, edgeBlockSum univ r c M t) / (M : ℂ)) / (N : ℂ)‖ ≤ |
| uniformAverage (fun t : Fin N => ‖edgeBlockSum univ r c M t.val‖) / M := by |
| simp only [norm_div, Complex.norm_natCast] |
| calc |
| _ ≤ ((∑ t ∈ range N, ‖edgeBlockSum univ r c M t‖) / (M : ℝ)) / N := by |
| gcongr |
| exact norm_sum_le _ _ |
| _ = _ := by |
| rw [uniformAverage, Fintype.card_fin, Fin.sum_univ_eq_sum_range |
| (fun t => ‖edgeBlockSum univ r c M t‖) N] |
| ring |
|
|
| theorem block_form_prefix_bound {E : Type*} [Fintype E] |
| (r : E → ℕ) (c : E → ℕ → ℂ) (A : ℕ → ℂ) |
| (M R N : ℕ) (hM : 0 < M) (hN : 0 < N) (L B D : ℝ) |
| (hL : 0 < L) (hD : 0 ≤ D) (hr : ∀ e, r e ≤ R) |
| (hrow : ∀ n : ℕ, 0 < n → (∑ e : E, ‖c e n‖) ≤ D) |
| (hA : ∀ t, A t = (2 * (L : ℂ)) * edgeBlockSum univ r c M t) |
| (havg : uniformAverage (fun t : Fin N => ‖A t.val‖) ≤ B) : |
| ‖positivePrefix (fun n => ∑ e : E, c e n) N / (N : ℂ)‖ ≤ |
| B / (2 * L * M) + (R : ℝ) / M * D + 2 * (M : ℝ) / N * D := by |
| have he : uniformAverage (fun t : Fin N => ‖A t.val‖) = |
| (2 * L) * uniformAverage (fun t : Fin N => ‖edgeBlockSum univ r c M t.val‖) := by |
| simp only [hA, norm_mul, Complex.norm_ofNat, |
| Complex.norm_real, Real.norm_eq_abs, abs_of_pos hL] |
| unfold uniformAverage |
| rw [← mul_sum] |
| ring |
| have hb : uniformAverage (fun t : Fin N => ‖edgeBlockSum univ r c M t.val‖) ≤ B / (2 * L) := by |
| apply (le_div_iff₀ (by positivity : 0 < 2 * L)).mpr |
| rw [mul_comm, ← he] |
| exact havg |
| have hprefix := edgeBlockSum_prefix_error univ r c M R N hM hN D hD |
| (fun e _ => hr e) hrow |
| let Z := ((∑ t ∈ range N, edgeBlockSum univ r c M t) / (M : ℂ)) / (N : ℂ) |
| have hZ : ‖Z‖ ≤ B / (2 * L * M) := by |
| apply (edgeBlockSum_norm_average_le r c M N).trans |
| calc |
| _ ≤ (B / (2 * L)) / M := div_le_div_of_nonneg_right hb (Nat.cast_nonneg _) |
| _ = _ := by ring |
| have ht := norm_sub_le Z |
| (Z - positivePrefix (fun n => ∑ e : E, c e n) N / (N : ℂ)) |
| have hid : Z - (Z - positivePrefix (fun n => ∑ e : E, c e n) N / (N : ℂ)) = |
| positivePrefix (fun n => ∑ e : E, c e n) N / (N : ℂ) := by ring |
| rw [hid] at ht |
| dsimp only [Z] at ht hZ |
| linarith |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
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|
