| import OAI.MeasureTheory.Falconer.Graphs.DistanceGraphPointwise |
| import OAI.MeasureTheory.Falconer.Measures.DistanceSchurIntegral |
| import OAI.MeasureTheory.Falconer.Graphs.ProductGraphEnergy |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ComplexConjugate |
|
|
| theorem original_distance_graph_integral_bound |
| (Q₁ Q₂ : DyadicSquare) (ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν] |
| (A B : CellPath) (N a n : ℕ) (sig t : ℝ) (hA : A.length = a) (hB : B.length = a) |
| {θ : ℝ} (hθ : 0 < θ) (hscale : θ = graphUncappedAngularScale N sig t) (axis : ℝ) (I : Finset GraphNonedgeIndex) |
| (hI : I ⊆ sourceParentGraphIndices N sig t (Q₁.descendant A) (Q₂.descendant B) ρ ν n A B) |
| {d : ℝ} (hd : 0 ≤ d) |
| (hdegree : ∀ j k s reverse row, |
| ((activeDistanceMatrixNeighbors I (a+n) j k θ axis s reverse row).card : ℝ) ≤ d) |
| {M₁ M₂ : ℝ} (hM₁ : 0 < M₁) (hM₂ : 0 < M₂) |
| (β γ : Plane → UnitCircle → ℂ) |
| (hβ : Measurable (fun z : Plane × UnitCircle => β z.1 z.2)) |
| (hγ : Measurable (fun z : Plane × UnitCircle => γ z.1 z.2)) |
| {B₁ B₂ : ℝ} (hB₁ : 0 ≤ B₁) (hB₂ : 0 ≤ B₂) |
| (hbβ : ∀ x u, ‖β x u‖ ≤ B₁) (hbγ : ∀ x u, ‖γ x u‖ ≤ B₂) |
| (v : ℝ) {Ψ : ℝ → ℝ} (hΨ : Measurable Ψ) (hpos : ∀ t, 0 ≤ Ψ t) |
| (hs : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4) |
| {W : ℝ} (hW : 0 ≤ W) (hw : ∀ t, Ψ t ≤ W) : |
| let U := fun P : CellPath => fun z : ℝ × UnitCircle => |
| sourceAngularTransform ρ (Q₁.descendant P).carrier M₁ β z.1 z.2 |
| let V := fun Q : CellPath => fun z : ℝ × UnitCircle => |
| sourceAngularTransform ν (Q₂.descendant Q).carrier M₂ γ z.1 z.2 |
| ‖((2:ℝ)^(2*(v-(a+n)))) • |
| ∑ e ∈ I, graphOriginalTerm ![ρ,ν,ν,ρ] |
| (fun l => (graphQuadCells ![Q₁,Q₂,Q₂,Q₁] e.1 l).carrier) |
| ![M₁,M₂,M₂,M₁] ![β,γ,γ,β] true e.2.2 axis θ v e.2.1.1 e.2.1.2 Ψ‖ ≤ |
| d * ∑ row ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n, |
| (sourceAngularH (a+n) v Ψ (U row.1) (V row.2)).toReal := by |
| classical |
| let : ∀ l : Fin 4, IsFiniteMeasure (![ρ,ν,ν,ρ] l) := by |
| intro l; fin_cases l <;> dsimp <;> infer_instance |
| have hM : ∀ l : Fin 4, 0 < (![M₁,M₂,M₂,M₁] l) := by |
| intro l; fin_cases l <;> first | exact hM₁ | exact hM₂ |
| have hb : ∀ l : Fin 4, Measurable (fun z : Plane × UnitCircle => (![β,γ,γ,β] l) z.1 z.2) := by |
| intro l; fin_cases l <;> first | exact hβ | exact hγ |
| have hbound : ∀ (l : Fin 4) x u, ‖(![β,γ,γ,β] l) x u‖ ≤ (![B₁,B₂,B₂,B₁] l) := by |
| intro l; fin_cases l <;> first | exact hbβ | exact hbγ |
| have hB4 : ∀ l : Fin 4, 0 ≤ (![B₁,B₂,B₂,B₁] l) := by |
| intro l; fin_cases l <;> first | exact hB₁ | exact hB₂ |
| let H := graphSupportedDistanceIndices I θ axis |
| let F := fun z : ℝ × (UnitCircle × UnitCircle) => ∑ e ∈ H, |
| sourceGraphFourTransform ![ρ,ν,ν,ρ] |
| (fun l => (graphQuadCells ![Q₁,Q₂,Q₂,Q₁] e.1 l).carrier) |
| ![M₁,M₂,M₂,M₁] ![β,γ,γ,β] |
| (graphRotatedDistanceWeight axis (angularBinCount θ) e.2.1.1 e.2.1.2) z.1 z.2.1 z.2.2 |
| let U := fun P : CellPath => fun z : ℝ × UnitCircle => |
| sourceAngularTransform ρ (Q₁.descendant P).carrier M₁ β z.1 z.2 |
| let V := fun Q : CellPath => fun z : ℝ × UnitCircle => |
| sourceAngularTransform ν (Q₂.descendant Q).carrier M₂ γ z.1 z.2 |
| have hpoint (r : ℝ) (u w : UnitCircle) : ‖F (r,(u,w))‖ ≤ |
| d/2 * ∑ row ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n, |
| (‖U row.1 (r,u)‖^2*‖V row.2 (r,w)‖^2 + ‖V row.2 (r,u)‖^2*‖U row.1 (r,w)‖^2) := |
| distance_graph_pointwise_bound Q₁ Q₂ ρ ν A B N a n sig t hA hB |
| I hI hθ hscale axis d hd hdegree (fun P u => U P (r,u)) (fun Q w => V Q (r,w)) u w |
| have hmain := original_distance_schur_integral (sourceProductVertices Q₁ Q₂ ρ ν A B n) |
| (fun row => U row.1) (fun row => V row.2) |
| (fun row _ => measurable_sourceAngularTransform ρ (Q₁.descendant row.1).carrier M₁ β hβ) |
| (fun row _ => measurable_sourceAngularTransform ν (Q₂.descendant row.2).carrier M₂ γ hγ) |
| (fun row => B₁*(ρ (Q₁.descendant row.1).carrier).toReal/M₁) |
| (fun row => B₂*(ν (Q₂.descendant row.2).carrier).toReal/M₂) |
| (fun row _ => by positivity) (fun row _ => by positivity) |
| (fun row _ z => sourceAngularTransform_norm_le ρ _ M₁ hM₁ β B₁ (fun x _ => hbβ x) z.1 z.2) |
| (fun row _ z => sourceAngularTransform_norm_le ν _ M₂ hM₂ γ B₂ (fun x _ => hbγ x) z.1 z.2) |
| F hd hpoint (a+n) v hΨ hpos hs hW hw |
| have hsum := sourceGraphFourTransform_original_sum H ![ρ,ν,ν,ρ] |
| (fun e l => (graphQuadCells ![Q₁,Q₂,Q₂,Q₁] e.1 l).carrier) |
| ![M₁,M₂,M₂,M₁] hM ![β,γ,γ,β] |
| (fun e => graphRotatedDistanceWeight axis (angularBinCount θ) e.2.1.1 e.2.1.2) hb |
| (fun e => (graphRotatedDistanceWeight_continuous _ _ _ _).measurable) |
| (fun e => graphRotatedDistanceWeight_norm_le _ _ _ _) |
| ![B₁,B₂,B₂,B₁] hB4 hbound hΨ hs hW |
| (fun t => by rw [abs_of_nonneg (hpos t)]; exact hw t) v |
| dsimp only [F] at hmain |
| rw [hsum] at hmain |
| dsimp only |
| rw [graphOriginalDistanceSum_supported] |
| simpa only [Finset.smul_sum,graphOriginalTerm,graphSourceWeight, |
| ↓reduceIte,H,U,V] using hmain |
|
|
| end PlanarFalconer |
|
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| end OAI |
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