File size: 3,722 Bytes
7f37411 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | 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
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