File size: 3,430 Bytes
7f37411 | 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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import OAI.MeasureTheory.Falconer.Geometry.DyadicCube
import Mathlib.Analysis.InnerProductSpace.Basic
namespace OAI
/-! # The actual distance Taylor error for the suffix reduction
The error estimate is dimension independent before insertion of the cube
diameter. Its proof is the exact squared-norm identity, so no differentiable
extension across the diagonal is needed.
-/
namespace Falconer
open Set
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
theorem norm_linearization_bounds (u v : E) (hu : 0 < ‖u‖) :
0 ≤ ‖v‖-inner ℝ (‖u‖⁻¹ • u) v ∧
‖v‖-inner ℝ (‖u‖⁻¹ • u) v ≤ ‖v-u‖^2/(2*‖u‖) := by
have hi : inner ℝ (‖u‖⁻¹ • u) v = inner ℝ u v/‖u‖ := by
rw [real_inner_smul_left]
ring
rw [hi]
have hb : inner ℝ u v/‖u‖ ≤ ‖v‖ := by
apply (div_le_iff₀ hu).mpr
exact (real_inner_le_norm u v).trans_eq (mul_comm _ _)
have he : 2*‖u‖*(‖v‖-inner ℝ u v/‖u‖) =
‖v-u‖^2-(‖v‖-‖u‖)^2 := by
have hs := norm_sub_sq_real v u
rw [real_inner_comm u v] at hs
field_simp
nlinarith
refine ⟨sub_nonneg.mpr hb,?_⟩
apply (le_div_iff₀ (mul_pos (by norm_num) hu)).mpr
nlinarith [sq_nonneg (‖v‖-‖u‖)]
theorem norm_linearization_abs_le (u v : E) (hu : 0 < ‖u‖) :
|‖v‖-inner ℝ (‖u‖⁻¹ • u) v| ≤ ‖v-u‖^2/(2*‖u‖) := by
rw [abs_of_nonneg (norm_linearization_bounds u v hu).1]
exact (norm_linearization_bounds u v hu).2
omit [InnerProductSpace ℝ E] in
theorem pair_displacement_bound (x y c₁ c₂ : E) {δ₁ δ₂ : ℝ}
(h₁ : ‖x-c₁‖ ≤ δ₁) (h₂ : ‖y-c₂‖ ≤ δ₂) :
‖(x-y)-(c₁-c₂)‖ ≤ δ₁+δ₂ := by
have he : (x-y)-(c₁-c₂) = (x-c₁)-(y-c₂) := by abel
rw [he]
exact (norm_sub_le _ _).trans (add_le_add h₁ h₂)
/-- Original distances on a pair of small cubes have the required
quadratic error relative to their center direction. -/
theorem DyadicCube.distance_linearization {d : ℕ} (P Q : DyadicCube d)
(hP : 0 < P.side) (hQ : Q.side = P.side)
{g : ℝ} (hg : 0 < g) (hsep : g ≤ ‖P.center-Q.center‖)
{x y : Space d} (hx : x ∈ P.carrier) (hy : y ∈ Q.carrier) :
|dist x y-inner ℝ (‖P.center-Q.center‖⁻¹ • (P.center-Q.center)) (x-y)| ≤
2*(d:ℝ)*P.side^2/g := by
have hQp : 0 < Q.side := by rw [hQ]; exact hP
have hδ : ‖(x-y)-(P.center-Q.center)‖ ≤ 2*Real.sqrt d*P.side := by
have h₁ := P.dist_le hP.le hx (P.center_mem hP)
have h₂ := Q.dist_le hQp.le hy (Q.center_mem hQp)
rw [hQ] at h₂
simpa only [dist_eq_norm,two_mul,mul_add,add_mul] using
pair_displacement_bound x y P.center Q.center h₁ h₂
have hsquare : ‖(x-y)-(P.center-Q.center)‖^2 ≤ 4*(d:ℝ)*P.side^2 := by
have hh := (sq_le_sq₀ (norm_nonneg _) (by positivity)).mpr hδ
nlinarith [Real.sq_sqrt (Nat.cast_nonneg (α := ℝ) d)]
have hnorm : 0 < ‖P.center-Q.center‖ := hg.trans_le hsep
rw [dist_eq_norm]
apply (norm_linearization_abs_le _ _ hnorm).trans
calc
_ ≤ (4*(d:ℝ)*P.side^2)/(2*‖P.center-Q.center‖) :=
div_le_div_of_nonneg_right hsquare (by positivity)
_ ≤ (4*(d:ℝ)*P.side^2)/(2*g) :=
div_le_div_of_nonneg_left (by positivity) (by positivity)
(mul_le_mul_of_nonneg_left hsep (by norm_num))
_ = _ := by ring
end Falconer
end OAI
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