File size: 2,766 Bytes
180bc03 | 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 | 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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