| import OAI.MeasureTheory.Falconer.Graphs.GraphCompleteWeightSupport |
| import OAI.MeasureTheory.Falconer.Graphs.GraphAmplitudeSupport |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set |
|
|
| theorem graph_directions_chartZ (φ z : ℝ) : |
| graphChartZ (sourceDirection (φ+Real.arctan z)) (sourceDirection (φ-Real.arctan z)) = z := by |
| let u := sourceDirection (φ+Real.arctan z) |
| let v := sourceDirection (φ-Real.arctan z) |
| have hn : 0 < ‖u+v‖ := by rw [graph_directions_sum_norm]; positivity |
| have hd := graph_chart_difference u v (norm_sourceDirection _) (norm_sourceDirection _) (norm_pos_iff.mp hn) |
| have he : u-v = z • quarterTurn (u+v) := by |
| dsimp [u,v] |
| rw [graph_directions_sum, quarterTurn_smul, sourceDirection_add_arctan, sourceDirection_sub_arctan] |
| module |
| have hq : quarterTurn (u+v) ≠ 0 := by rw [← norm_ne_zero_iff,graph_quarterTurn_norm]; exact hn.ne' |
| exact (smul_left_injective ℝ hq) (hd.symm.trans he) |
|
|
| theorem graph_directions_chartAxis (φ z : ℝ) : |
| graphChartAxis (sourceDirection (φ+Real.arctan z)) (sourceDirection (φ-Real.arctan z)) = sourceDirection φ := by |
| rw [graphChartAxis, graph_directions_sum_norm, graph_directions_sum, smul_smul, |
| inv_mul_cancel₀ (by positivity : (2:ℝ)/Real.sqrt (1+z^2) ≠ 0), one_smul] |
|
|
| noncomputable def graphSourceEndpoints (distance plus : Bool) (p : Plane × ℝ) : UnitCircle × UnitCircle := |
| (sourceUnitDirection (graphPlaneAngle p.1+Real.arctan p.2), |
| sourceUnitDirection (graphPlaneAngle p.1-Real.arctan p.2+graphSourceChartAxis distance plus)) |
|
|
| theorem graphSourceEndpoints_signed (distance plus : Bool) (p : Plane × ℝ) : |
| graphSourceSign distance plus • ((graphSourceEndpoints distance plus p).2:Plane) = |
| sourceDirection (graphPlaneAngle p.1-Real.arctan p.2) := by |
| cases distance <;> cases plus <;> |
| simp [graphSourceEndpoints,graphSourceSign,graphSourceChartAxis,sourceUnitDirection,sourceDirection_add_pi] |
|
|
| theorem graphSourceEndpoints_coordinates (distance plus : Bool) {p : Plane × ℝ} (hp : p.1 ≠ 0) : |
| graphChartZ ((graphSourceEndpoints distance plus p).1:Plane) |
| (graphSourceSign distance plus • ((graphSourceEndpoints distance plus p).2:Plane)) = p.2 ∧ |
| graphChartAxis ((graphSourceEndpoints distance plus p).1:Plane) |
| (graphSourceSign distance plus • ((graphSourceEndpoints distance plus p).2:Plane)) = graphNormalizedAxis p.1 := by |
| rw [graphSourceEndpoints_signed] |
| change graphChartZ (sourceDirection _) _ = _ ∧ graphChartAxis (sourceDirection _) _ = _ |
| rw [graph_directions_chartZ, graph_directions_chartAxis,graphNormalizedAxis_eq_all hp] |
| exact ⟨rfl,rfl⟩ |
|
|
| theorem graphAmplitudePullback_weight_nonzero (b : Fin 4 → Plane → UnitCircle → ℂ) |
| (x : Fin 4 → Plane) (distance plus : Bool) (axis δ : ℝ) (j k : ℕ) |
| (Ψ : ℝ → ℝ) {p : Plane × ℝ} |
| (hp : graphAmplitudePullback b x distance plus axis δ j k Ψ p ≠ 0) : |
| graphSourceWeight distance plus axis (angularBinCount δ) j k (graphSourceEndpoints distance plus p) ≠ 0 := by |
| intro hz |
| apply hp |
| change graphSourceWeight distance plus axis (angularBinCount δ) j k |
| (sourceUnitDirection (graphPlaneAngle p.1+Real.arctan p.2), |
| sourceUnitDirection (graphPlaneAngle p.1-Real.arctan p.2+graphSourceChartAxis distance plus)) = 0 at hz |
| simp [graphAmplitudePullback,graphFourAmplitude,hz] |
|
|
| def GraphSourceRepresentative (distance plus : Bool) (axis δ : ℝ) (j k : ℕ) |
| (q : UnitCircle × UnitCircle) : Prop := |
| q ∈ tsupport (graphSourceWeight distance plus axis (angularBinCount δ) j k) ∧ |
| ∀ w ∈ tsupport (graphSourceWeight distance plus axis (angularBinCount δ) j k), |
| |graphChartZ (w.1:Plane) (graphSourceSign distance plus • (w.2:Plane))- |
| graphChartZ (q.1:Plane) (graphSourceSign distance plus • (q.2:Plane))|+ |
| dist (graphChartAxis (w.1:Plane) (graphSourceSign distance plus • (w.2:Plane))) |
| (graphChartAxis (q.1:Plane) (graphSourceSign distance plus • (q.2:Plane))) ≤ 112*δ |
|
|
| theorem graphSourceRepresentative_exists {δ : ℝ} (hδ : 0 < δ) (hcap : δ ≤ 2*Real.pi) |
| (distance plus : Bool) (axis : ℝ) (j k : ℕ) |
| (hS : (tsupport (graphSourceWeight distance plus axis (angularBinCount δ) j k)).Nonempty) : |
| ∃ q, GraphSourceRepresentative distance plus axis δ j k q := |
| graphSourceWeight_complete_representative hδ hcap distance plus axis j k hS |
|
|
| theorem GraphSourceRepresentative.amplitude_coordinates |
| {distance plus : Bool} {axis δ : ℝ} {j k : ℕ} {q : UnitCircle × UnitCircle} |
| (hq : GraphSourceRepresentative distance plus axis δ j k q) |
| (b : Fin 4 → Plane → UnitCircle → ℂ) (x : Fin 4 → Plane) |
| {Ψ : ℝ → ℝ} (hsupp : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4) |
| {p : Plane × ℝ} (hp : graphAmplitudePullback b x distance plus axis δ j k Ψ p ≠ 0) : |
| |p.2-graphChartZ (q.1:Plane) (graphSourceSign distance plus • (q.2:Plane))|+ |
| dist (graphNormalizedAxis p.1) (graphChartAxis (q.1:Plane) (graphSourceSign distance plus • (q.2:Plane))) ≤ 112*δ := by |
| have hs := graphAmplitudePullback_tsupport b x distance plus axis δ j k hsupp (subset_closure hp) |
| have hlo : (1:ℝ)/8 ≤ ‖p.1‖ := hs.2 |
| have hn : p.1 ≠ 0 := norm_pos_iff.mp (by linarith) |
| have hw := hq.2 _ (subset_closure (graphAmplitudePullback_weight_nonzero b x distance plus axis δ j k Ψ hp)) |
| rw [(graphSourceEndpoints_coordinates distance plus hn).1, |
| (graphSourceEndpoints_coordinates distance plus hn).2] at hw |
| exact hw |
|
|
| theorem graphAmplitudePullback_zero_of_empty_weight_support |
| (b : Fin 4 → Plane → UnitCircle → ℂ) (x : Fin 4 → Plane) |
| (distance plus : Bool) (axis δ : ℝ) (j k : ℕ) (Ψ : ℝ → ℝ) |
| (hS : ¬(tsupport (graphSourceWeight distance plus axis (angularBinCount δ) j k)).Nonempty) : |
| graphAmplitudePullback b x distance plus axis δ j k Ψ = 0 := by |
| funext p |
| by_contra hp |
| exact hS ⟨_,subset_closure (graphAmplitudePullback_weight_nonzero b x distance plus axis δ j k Ψ hp)⟩ |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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