| import OAI.MeasureTheory.Falconer.Angular.AngularChart | |
| namespace OAI | |
| /-! | |
| # The failed perpendicular test in an actual spherical chart | |
| The normalized affine chart stays within twice its tangent displacement | |
| of the cap center. Its coordinate derivative therefore inherits the | |
| uniform lower bound proved for the tangential sphere field. | |
| -/ | |
| namespace Falconer | |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] | |
| theorem norm_normalization_sub_le {w z : E} (hw : ‖w‖ = 1) (hz : z ≠ 0) : | |
| ‖‖z‖⁻¹ • z - w‖ ≤ 2*‖z-w‖ := by | |
| have hzn : ‖z‖ ≠ 0 := norm_ne_zero_iff.mpr hz | |
| have he : ‖z‖⁻¹ • z - w = (z-w) + (‖z‖⁻¹-1) • z := by module | |
| have hnorm : ‖(‖z‖⁻¹-1) • z‖ = |‖z‖-1| := by | |
| rw [norm_smul,Real.norm_eq_abs] | |
| calc | |
| _ = |(‖z‖⁻¹-1)*‖z‖| := by rw [abs_mul,abs_of_nonneg (norm_nonneg z)] | |
| _ = |1-‖z‖| := by rw [sub_mul,inv_mul_cancel₀ hzn,one_mul] | |
| _ = _ := abs_sub_comm _ _ | |
| rw [he] | |
| calc | |
| _ ≤ ‖z-w‖ + ‖(‖z‖⁻¹-1) • z‖ := norm_add_le _ _ | |
| _ = ‖z-w‖ + |‖z‖-1| := by rw [hnorm] | |
| _ ≤ ‖z-w‖ + ‖z-w‖ := by | |
| exact add_le_add le_rfl (by simpa only [hw] using abs_norm_sub_norm_le z w) | |
| _ = _ := by ring | |
| theorem tangent_chart_nonzero {w h : E} (hw : ‖w‖ = 1) (hh : inner ℝ w h = 0) : | |
| w+h ≠ 0 := by | |
| intro he | |
| have hi := congrArg (inner ℝ w) he | |
| simp only [inner_add_right,real_inner_self_eq_norm_sq,hw,one_pow,hh,add_zero, | |
| inner_zero_right] at hi | |
| norm_num at hi | |
| /-- A single coordinate direction works for every allowed point and | |
| direction replacement in the discarded cell--cap term. -/ | |
| theorem perpendicular_failure_chart_nonstationary {w₀ d : E} | |
| (hw₀ : ‖w₀‖ = 1) {τ : ℝ} (hτ : 0 < τ) | |
| (hbad : τ < ‖perpendicularPart w₀ d‖) : | |
| ∃ t : E, ‖t‖ = 1 ∧ inner ℝ w₀ t = 0 ∧ | |
| ∀ (y v : E) (u : ℝ), inner ℝ w₀ y = 0 → ‖y+u•t‖ ≤ 1 → | |
| ‖v-d‖ + 4*‖y+u•t‖*‖d‖ ≤ τ/2 → | |
| τ/4 ≤ |deriv (fun x => inner ℝ (normalizedLine (w₀+y) t x) v) u| := by | |
| obtain ⟨t,ht,htan,hfield⟩ := perpendicular_failure_nonstationary hw₀ hτ hbad | |
| refine ⟨t,ht,htan,?_⟩ | |
| intro y v u hy hsmall herr | |
| have hh : inner ℝ w₀ (y+u•t) = 0 := by | |
| simp only [inner_add_right,real_inner_smul_right,hy,htan,mul_zero,add_zero] | |
| have hne : (w₀+y)+u•t ≠ 0 := by | |
| rw [add_assoc] | |
| exact tangent_chart_nonzero hw₀ hh | |
| have hnorm : ‖(w₀+y)+u•t‖ ≤ 2 := by | |
| rw [add_assoc] | |
| exact (norm_add_le _ _).trans (by rw [hw₀]; linarith) | |
| have hnear : ‖normalizedLine (w₀+y) t u - w₀‖ ≤ 2*‖y+u•t‖ := by | |
| have hp := norm_normalization_sub_le hw₀ hne | |
| simpa only [normalizedLine,add_assoc,add_sub_cancel_left] using hp | |
| have hfield' := hfield (normalizedLine (w₀+y) t u) v | |
| (normalizedLine_norm _ _ hne) (by | |
| have hm := mul_le_mul_of_nonneg_right hnear (norm_nonneg d) | |
| nlinarith) | |
| exact normalizedLine_phase_nonstationary _ _ _ hne hnorm hτ.le hfield' | |
| end Falconer | |
| end OAI | |