| import OAI.MeasureTheory.Falconer.Geometry.MaskedGeometry |
| import OAI.MeasureTheory.Falconer.Angular.NorthRotation |
| import OAI.MeasureTheory.Falconer.Radial.RadialFreezing |
|
|
| namespace OAI |
|
|
| /-! # One angular frame for all pairs in separated original cells -/ |
|
|
| namespace Falconer |
| open Set MeasureTheory NormedSpace |
|
|
| theorem separated_cells_north_frame (m : ℕ) (P Q : DyadicCube (m+3)) |
| (hP : 0 < P.side) (hQ : Q.side = P.side) |
| (hsep : 125*(((m+3:ℕ):ℝ)+1)*P.side ≤ ‖P.center-Q.center‖) : |
| ∃ T : Space (m+3) ≃ₗᵢ[ℝ] Space (m+3), |
| ∀ x ∈ P.carrier, ∀ y ∈ Q.carrier, |
| ‖T (normalize (x-y))-(gnomonicChart (0 : Space (m+2)) : Space (m+3))‖ ≤ 1/8 := by |
| have hs : 0 < 125*(((m+3:ℕ):ℝ)+1)*P.side := by positivity |
| have hc : P.center-Q.center ≠ 0 := norm_pos_iff.mp (hs.trans_le hsep) |
| let u : Sphere (m+3) := ⟨normalize (P.center-Q.center),by |
| rw [Metric.mem_sphere,dist_zero_right,norm_normalize hc]⟩ |
| refine ⟨northRotation u,?_⟩ |
| intro x hx y hy |
| have hpoints := P.separated_pairs_lower Q hP hQ hsep hx hy |
| have hxy : x-y ≠ 0 := norm_pos_iff.mp |
| ((show 0 < 100*(((m+3:ℕ):ℝ)+1)*P.side by positivity).trans_le hpoints) |
| have hx' := P.dist_le hP.le (P.center_mem hP) hx |
| have hy' := Q.dist_le (by rw [hQ]; exact hP.le) hy |
| (Q.center_mem (by rw [hQ]; exact hP)) |
| rw [hQ] at hy' |
| have hd : dist (P.center-Q.center) (x-y) ≤ 2*Real.sqrt (m+3:ℕ)*P.side := by |
| rw [dist_eq_norm,show (P.center-Q.center)-(x-y) = (P.center-x)+(y-Q.center) by abel] |
| exact (norm_add_le _ _).trans (by rw [← dist_eq_norm,← dist_eq_norm]; linarith) |
| have hn := normalize_distance_scaled hsep hxy |
| have hsmall : dist (normalize (x-y)) (normalize (P.center-Q.center)) ≤ 1/8 := by |
| rw [dist_comm] at hn |
| have hdim := mul_le_mul_of_nonneg_right (sqrt_dimension_le (m+3)) hP.le |
| nlinarith |
| rw [← linearIsometryEquiv_normalize_sub,northRotation_cap_norm] |
| exact hsmall |
|
|
| end Falconer |
|
|
| end OAI |
|
|
