| import OAI.MeasureTheory.Falconer.Angular.AngularStripMeasure |
| import Mathlib.MeasureTheory.Integral.Lebesgue.Markov |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ENNReal |
|
|
| noncomputable def stripHitBound (W : ℝ) (z : Plane) : ℝ≥0∞ := |
| if z=0 then ENNReal.ofReal (2*Real.pi) |
| else ENNReal.ofReal (4*Real.pi*min 1 (W/‖z‖)) |
|
|
| theorem sourceArcLength_strip_le_bound (z : Plane) {W : ℝ} (hW : 0 ≤ W) : |
| sourceArcLength {ω : UnitCircle | |planeDot z (ω:Plane)| ≤ W} ≤ stripHitBound W z := by |
| by_cases hz : z=0 |
| · simp only [stripHitBound,ite_eq_left hz] |
| exact (measure_mono (subset_univ _)).trans_eq sourceArcLength_univ |
| · simpa only [stripHitBound,ite_eq_right hz] using sourceArcLength_strip z hz hW |
|
|
| theorem stripCount_lintegral {ι : Type*} (S : Finset ι) (v : ι → Plane) (W : ℝ) : |
| (∫⁻ ω : UnitCircle, (stripCount S v W (ω:Plane):ℝ≥0∞) ∂sourceArcLength) = |
| ∑ i ∈ S, sourceArcLength {ω : UnitCircle | |planeDot (v i) (ω:Plane)| ≤ W} := by |
| classical |
| let f : ι → UnitCircle → ℝ≥0∞ := fun i => |
| {ω : UnitCircle | |planeDot (v i) (ω:Plane)| ≤ W}.indicator (fun _ => 1) |
| have he (ω : UnitCircle) : (stripCount S v W (ω:Plane):ℝ≥0∞) = ∑ i ∈ S, f i ω := by |
| rw [stripCount,Finset.card_filter,Nat.cast_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| by_cases hi : |planeDot (v i) (ω:Plane)| ≤ W <;> simp [hi,f] |
| simp_rw [he] |
| rw [lintegral_finsetSum S (fun i _ => measurable_const.indicator (measurableSet_directionStrip (v i) W))] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [lintegral_indicator (measurableSet_directionStrip (v i) W)] |
| simp |
|
|
| theorem stripCount_mean_bound {ι : Type*} (S : Finset ι) (v : ι → Plane) {W : ℝ} (hW : 0 ≤ W) : |
| (∫⁻ ω : UnitCircle, (stripCount S v W (ω:Plane):ℝ≥0∞) ∂sourceArcLength) ≤ |
| ∑ i ∈ S, stripHitBound W (v i) := by |
| rw [stripCount_lintegral] |
| exact Finset.sum_le_sum (fun i _ => sourceArcLength_strip_le_bound (v i) hW) |
|
|
| theorem stripCount_exceptional_bound {ι : Type*} (S : Finset ι) (v : ι → Plane) |
| {W T : ℝ} (hW : 0 ≤ W) (hT : 0 < T) : |
| sourceArcLength {ω : UnitCircle | T < (stripCount S v W (ω:Plane):ℝ)} ≤ |
| (∑ i ∈ S, stripHitBound W (v i))/ENNReal.ofReal T := by |
| have hm : Measurable (fun ω : UnitCircle => (stripCount S v W (ω:Plane):ℝ≥0∞)) := |
| (measurable_of_countable (fun n : ℕ => (n:ℝ≥0∞))).comp |
| ((measurable_stripCount S v W).comp continuous_subtype_val.measurable) |
| have hsub : {ω : UnitCircle | T < (stripCount S v W (ω:Plane):ℝ)} ⊆ |
| {ω : UnitCircle | ENNReal.ofReal T ≤ (stripCount S v W (ω:Plane):ℝ≥0∞)} := by |
| intro ω hω |
| simpa using ENNReal.ofReal_le_ofReal hω.le |
| apply (measure_mono hsub).trans |
| apply (meas_ge_le_lintegral_div hm.aemeasurable |
| (by positivity : ENNReal.ofReal T ≠ 0) ENNReal.ofReal_ne_top).trans |
| gcongr |
| exact stripCount_mean_bound S v hW |
|
|
| theorem planeDot_quarterTurn (z ω : Plane) : |
| planeDot z (quarterTurn ω) = -planeDot (quarterTurn z) ω := by |
| simp [planeDot,quarterTurn] |
|
|
| theorem parallelTestCount_as_strip (Q : DyadicSquare) (ρ : Measure Plane) |
| (g p : ℕ) (P : CellPath) (W : ℝ) (ω : Plane) : |
| parallelTestCount Q ρ g p P W ω = |
| stripCount (parallelTestCells Q ρ g p P) |
| (fun c => quarterTurn ((Q.descendant c).center-(Q.descendant P).center)) W ω := by |
| classical |
| simp only [parallelTestCount,stripCount,planeDot_quarterTurn,abs_neg] |
|
|
| theorem parallelGoodSet_compl_bound (Q : DyadicSquare) (ρ : Measure Plane) |
| (g p : ℕ) (P : CellPath) {W T : ℝ} (hW : 0 ≤ W) (hT : 0 < T) : |
| sourceArcLength (parallelGoodSet Q ρ g p P W T)ᶜ ≤ |
| (∑ c ∈ parallelTestCells Q ρ g p P, |
| stripHitBound W (quarterTurn ((Q.descendant c).center-(Q.descendant P).center)))/ |
| ENNReal.ofReal T := by |
| have he : (parallelGoodSet Q ρ g p P W T)ᶜ = |
| {ω : UnitCircle | T < (stripCount (parallelTestCells Q ρ g p P) |
| (fun c => quarterTurn ((Q.descendant c).center-(Q.descendant P).center)) W (ω:Plane):ℝ)} := by |
| ext ω |
| simp [parallelGoodSet,parallelTestCount_as_strip] |
| rw [he] |
| exact stripCount_exceptional_bound _ _ hW hT |
|
|
| theorem projectionGoodSet_compl_bound (Q : DyadicSquare) (ρ : Measure Plane) |
| (p u : ℕ) (P : CellPath) {W T : ℝ} (hW : 0 ≤ W) (hT : 0 < T) : |
| sourceArcLength (projectionGoodSet Q ρ p u P W T)ᶜ ≤ |
| (∑ cc ∈ (projectionTestCells Q ρ p u P).product (projectionTestCells Q ρ p u P), |
| stripHitBound W ((Q.descendant cc.1).center-(Q.descendant cc.2).center))/ |
| ENNReal.ofReal T := by |
| have he : (projectionGoodSet Q ρ p u P W T)ᶜ = |
| {ω : UnitCircle | T < (stripCount |
| ((projectionTestCells Q ρ p u P).product (projectionTestCells Q ρ p u P)) |
| (fun cc => (Q.descendant cc.1).center-(Q.descendant cc.2).center) W (ω:Plane):ℝ)} := by |
| ext ω |
| simp [projectionGoodSet,projectionTestCount] |
| rw [he] |
| exact stripCount_exceptional_bound _ _ hW hT |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
