| import OAI.MeasureTheory.Falconer.Graphs.GraphGlobalPhase |
| import OAI.MeasureTheory.Falconer.Graphs.GraphChartRegularity |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set Metric |
| open scoped ContDiff |
|
|
| noncomputable def graphNormalizedAxis (η : Plane) : Plane := ‖η‖⁻¹ • η |
|
|
| noncomputable def graphAlgebraicDirection (plus : Bool) (p : Plane × ℝ) : Plane := |
| (Real.sqrt (1+p.2^2))⁻¹ • |
| (graphNormalizedAxis p.1 + (if plus then p.2 else -p.2) • quarterTurn (graphNormalizedAxis p.1)) |
|
|
| theorem graphNormalizedAxis_contDiffAt (η : Plane) (hη : η ≠ 0) : |
| ContDiffAt ℝ ∞ graphNormalizedAxis η := |
| ((contDiffAt_id.norm ℝ hη).inv (norm_ne_zero_iff.mpr hη)).smul contDiffAt_id |
|
|
| theorem quarterTurn_contDiff : ContDiff ℝ ∞ quarterTurn := by |
| apply (contDiff_piLp 2).mpr |
| intro i |
| fin_cases i |
| · change ContDiff ℝ ∞ (fun x : Plane => -x 1) |
| fun_prop |
| · change ContDiff ℝ ∞ (fun x : Plane => x 0) |
| fun_prop |
|
|
| theorem graphAlgebraicDirection_contDiffAt (plus : Bool) (p : Plane × ℝ) (hp : p.1 ≠ 0) : |
| ContDiffAt ℝ ∞ (graphAlgebraicDirection plus) p := by |
| have ha : ContDiffAt ℝ ∞ (fun q : Plane × ℝ => graphNormalizedAxis q.1) p := |
| (graphNormalizedAxis_contDiffAt p.1 hp).comp p contDiffAt_fst |
| have hs : ContDiffAt ℝ ∞ (fun q : Plane × ℝ => (Real.sqrt (1+q.2^2))⁻¹) p := |
| ((contDiffAt_const.add (contDiffAt_snd.pow 2)).sqrt (by positivity)).inv (by positivity) |
| have hz : ContDiffAt ℝ ∞ (fun q : Plane × ℝ => if plus then q.2 else -q.2) p := by |
| cases plus <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> fun_prop |
| exact hs.smul (ha.add (hz.smul (quarterTurn_contDiff.contDiffAt.comp p ha))) |
|
|
| theorem graphNormalizedAxis_eq {η : Plane} (hη : η ∈ graphPlaneSlit) : |
| graphNormalizedAxis η = sourceDirection (graphPlaneAngle η) := by |
| have hp := polarCoord.map_source hη |
| have hn : 0 < ‖η‖ := by |
| simpa only [graphPlaneEquiv_symm_radius, mem_Ioi] using hp.1 |
| calc |
| graphNormalizedAxis η = ‖η‖⁻¹ • (‖η‖ • sourceDirection (graphPlaneAngle η)) := |
| congrArg (fun v : Plane => ‖η‖⁻¹ • v) (graph_plane_axis hη).symm |
| _ = _ := by rw [smul_smul, inv_mul_cancel₀ hn.ne', one_smul] |
| |
| theorem graphAlgebraicDirection_eq (plus : Bool) {η : Plane} (hη : η ∈ graphPlaneSlit) (z : ℝ) : |
| graphAlgebraicDirection plus (η,z) = |
| sourceDirection (graphPlaneAngle η+(if plus then Real.arctan z else -Real.arctan z)) := by |
| cases plus |
| · simp only [graphAlgebraicDirection, Bool.false_eq_true, ↓reduceIte, |
| graphNormalizedAxis_eq hη, neg_smul, ← sub_eq_add_neg] |
| exact (sourceDirection_sub_arctan _ _).symm |
| · simp only [graphAlgebraicDirection, ↓reduceIte, graphNormalizedAxis_eq hη] |
| exact (sourceDirection_add_arctan _ _).symm |
| |
| theorem graphAlgebraicDirection_uniform_derivatives (n : ℕ) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ (plus : Bool) (η : Plane) (z : ℝ), |
| ‖η‖ ≤ 8 → |z| ≤ 4 → 1/8 ≤ ‖η‖ → |
| ‖iteratedFDeriv ℝ n (graphAlgebraicDirection plus) (η,z)‖ ≤ C := by |
| let K : Set (Plane × ℝ) := closedBall 0 8 ∩ {p | 1/8 ≤ ‖p.1‖} |
| have hK : IsCompact K := (isCompact_closedBall (0:Plane × ℝ) 8).inter_right |
| (isClosed_le continuous_const continuous_fst.norm) |
| have hc (plus : Bool) : ContinuousOn (iteratedFDeriv ℝ n (graphAlgebraicDirection plus)) K := by |
| intro p hp |
| have hr : 1/8 ≤ ‖p.1‖ := hp.2 |
| have hn : p.1 ≠ 0 := norm_pos_iff.mp (by linarith) |
| exact ((graphAlgebraicDirection_contDiffAt plus p hn).continuousAt_iteratedFDeriv |
| (by exact_mod_cast (le_top : (n : ℕ∞) ≤ ⊤))).continuousWithinAt |
| obtain ⟨A,hA⟩ := hK.exists_bound_of_continuousOn (hc true) |
| obtain ⟨B,hB⟩ := hK.exists_bound_of_continuousOn (hc false) |
| refine ⟨max 0 (max A B), le_max_left _ _, ?_⟩ |
| intro plus η z hη hz hlo |
| have hp : (η,z) ∈ K := by |
| refine ⟨?_,hlo⟩ |
| simp only [mem_closedBall, dist_zero_right, Prod.norm_def, max_le_iff, Real.norm_eq_abs] |
| exact ⟨hη,hz.trans (by norm_num)⟩ |
| cases plus |
| · exact (hB _ hp).trans ((le_max_right A B).trans (le_max_right _ _)) |
| · exact (hA _ hp).trans ((le_max_left A B).trans (le_max_right _ _)) |
| |
| end PlanarFalconer |
| |
| end OAI |
| |
