| import OAI.MeasureTheory.Falconer.Recursion.ParentConeSource |
| import OAI.MeasureTheory.Falconer.Angular.SmoothRotatedCone |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| theorem sourceConeCircle_zero_of_cos_neg (axis θ : ℝ) |
| (hc : Real.cos (θ-axis) < 0) : sourceConeCircle axis (sourceUnitDirection θ) = 0 := by |
| by_contra hn |
| have hs : sourceAngle (sourceUnitDirection (θ-axis)) ∈ Icc (-1/5:ℝ) (1/5) := by |
| apply graphConeCutoff_support |
| apply subset_closure |
| simpa only [Function.mem_support,sourceConeCircle,sourceCircleRotate_comp, |
| sub_eq_add_neg] using hn |
| have hcos : 0 ≤ Real.cos (sourceAngle (sourceUnitDirection (θ-axis))) := |
| Real.cos_nonneg_of_neg_pi_div_two_le_of_le |
| (by linarith only [hs.1, Real.one_le_pi_div_two]) |
| (by linarith only [hs.2, Real.one_le_pi_div_two]) |
| rw [periodic_sourceAngle Real.cos_periodic] at hcos |
| exact (not_lt_of_ge hcos) hc |
|
|
| theorem sourceConeCircle_inverse_germs (axis : ℝ) {φ : ℝ} (hφ : |φ| ≤ 1/20) : |
| (fun θ => sourceConeCircle axis (sourceUnitDirection θ)) =ᶠ[𝓝 (φ+axis)] (fun _ => 1) ∧ |
| (fun θ => sourceConeCircle axis (sourceUnitDirection θ)) =ᶠ[𝓝 (φ+axis+Real.pi)] (fun _ => 0) := by |
| constructor |
| · filter_upwards [Metric.ball_mem_nhds (φ+axis) (by norm_num : (0:ℝ) < 1/20)] with θ hθ |
| have hh : |(θ-axis)-φ| ≤ 1/20 := by |
| rw [Metric.mem_ball,Real.dist_eq] at hθ |
| convert hθ.le using 1 |
| congr 1 |
| ring |
| simpa using sourceConeCircle_one_near_centered_angle axis hφ hh |
| · have hc : Real.cos (φ+axis+Real.pi-axis) < 0 := by |
| have hp : -(Real.pi/2) < φ ∧ φ < Real.pi/2 := by |
| have hh := abs_le.mp hφ |
| constructor |
| · linarith only [hh.1, Real.pi_gt_three] |
| · linarith only [hh.2, Real.pi_gt_three] |
| rw [show φ+axis+Real.pi-axis = φ+Real.pi by ring,Real.cos_add_pi] |
| exact neg_neg_of_pos (Real.cos_pos_of_mem_Ioo hp) |
| have hne : {θ : ℝ | Real.cos (θ-axis) < 0} ∈ 𝓝 (φ+axis+Real.pi) := |
| (isOpen_lt (Real.continuous_cos.comp (continuous_id.sub continuous_const)) |
| continuous_const).mem_nhds hc |
| filter_upwards [hne] with θ hθ |
| exact sourceConeCircle_zero_of_cos_neg axis θ hθ |
|
|
| theorem sourceConeCircle_uniform_derivatives (n : ℕ) : |
| ∃ P : ℝ, 0 ≤ P ∧ ∀ axis : ℝ, ∀ j ≤ n, ∀ θ, |
| ‖iteratedDeriv j (fun t => sourceConeCircle axis (sourceUnitDirection t)) θ‖ ≤ P := by |
| let f := fun θ => sourceConeCircle 0 (sourceUnitDirection θ) |
| have hf : ContDiff ℝ ∞ f := sourceConeCircle_angular_contDiff |
| have hp : Function.Periodic f (2*Real.pi) := |
| fun θ => congrArg (sourceConeCircle 0) (sourceUnitDirection_periodic θ) |
| have hj (j : ℕ) : ∃ B : ℝ, 0 ≤ B ∧ ∀ θ, ‖iteratedDeriv j f θ‖ ≤ B := by |
| have hpj : Function.Periodic (iteratedDeriv j f) (2*Real.pi) := by |
| have he : (fun x => f (x+2*Real.pi)) = f := funext hp |
| have hh := iteratedDeriv_comp_add_const j f (2*Real.pi) |
| rw [he] at hh |
| intro x |
| exact (congrFun hh x).symm |
| obtain ⟨B,hB⟩ := (hpj.isBounded_of_continuous |
| (by positivity : 2*Real.pi ≠ 0) (hf.continuous_iteratedDeriv j |
| (by exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤)))).exists_norm_le |
| exact ⟨max 0 B,le_max_left _ _,fun θ => (hB _ ⟨θ,rfl⟩).trans (le_max_right _ _)⟩ |
| choose B hB hb using hj |
| refine ⟨∑ j ∈ Finset.range (n+1), B j,Finset.sum_nonneg (fun j _ => hB j),?_⟩ |
| intro axis j hj θ |
| rw [sourceConeCircle_rotated_deriv] |
| exact (hb j (θ-axis)).trans (Finset.single_le_sum (fun k _ => hB k) |
| (Finset.mem_range.mpr (Nat.lt_succ_of_le hj))) |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
