| 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) |
|
|
| / |
| 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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|
