| import OAI.MeasureTheory.Falconer.Estimates.DistanceOccupiedSum |
| import OAI.MeasureTheory.Falconer.Estimates.DistanceWeightSum |
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|
| namespace OAI |
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|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ComplexConjugate |
|
|
| theorem distance_graph_pointwise_bound |
| (Q₁ Q₂ : DyadicSquare) (ρ ν : Measure Plane) (A B : CellPath) (N a n : ℕ) |
| (sig t : ℝ) (hA : A.length = a) (hB : B.length = a) |
| (I : Finset GraphNonedgeIndex) |
| (hI : I ⊆ sourceParentGraphIndices N sig t (Q₁.descendant A) (Q₂.descendant B) ρ ν n A B) |
| {θ : ℝ} (hθ : 0 < θ) (hscale : θ = graphUncappedAngularScale N sig t) |
| (axis 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) |
| (U V : CellPath → UnitCircle → ℂ) (u w : UnitCircle) : |
| ‖∑ e ∈ graphSupportedDistanceIndices I θ axis, |
| graphRotatedDistanceWeight axis (angularBinCount θ) e.2.1.1 e.2.1.2 (u,w) * |
| (U e.1.1.1 u*conj (V e.1.1.2 u))*(V e.1.2.1 w*conj (U e.1.2.2 w))‖ ≤ |
| d/2 * ∑ row ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n, |
| (‖U row.1 u‖^2*‖V row.2 w‖^2 + ‖V row.2 u‖^2*‖U row.1 w‖^2) := by |
| classical |
| let C := sourceProductVertices Q₁ Q₂ ρ ν A B n |
| let D := sourceProductVertices Q₂ Q₁ ν ρ B A n |
| let H := graphSupportedDistanceIndices I θ axis |
| let F := fun e : GraphNonedgeIndex => graphRotatedDistanceWeight axis (angularBinCount θ) e.2.1.1 e.2.1.2 (u,w) * |
| (U e.1.1.1 u*conj (V e.1.1.2 u))*(V e.1.2.1 w*conj (U e.1.2.2 w)) |
| let S := d/2 * ((∑ row ∈ C, ‖U row.1 u‖^2*‖V row.2 w‖^2) + |
| (∑ col ∈ D, ‖V col.1 u‖^2*‖U col.2 w‖^2)) |
| have hS : 0 ≤ S := by dsimp [S]; positivity |
| have hh : H ⊆ sourceParentGraphIndices N sig t (Q₁.descendant A) (Q₂.descendant B) ρ ν n A B := |
| (Finset.filter_subset _ _).trans hI |
| have hmatrix (j b : ℕ) : |
| ‖∑ row ∈ C, ∑ col ∈ D, if distanceMatrixIndex row col j b true ∈ H |
| then F (distanceMatrixIndex row col j b true) else 0‖ ≤ |
| ‖graphRotatedDistanceWeight axis (angularBinCount θ) j b (u,w)‖ * S := by |
| have he := distance_matrix_complex_bound_on I (a+n) j b θ axis true C D |
| (sourceProductVertices_subset Q₁ Q₂ ρ ν A B a n hA hB) |
| (sourceProductVertices_subset Q₂ Q₁ ν ρ B A a n hB hA) |
| (fun row => U row.1 u*V row.2 w) (fun col => V col.1 u*U col.2 w) |
| (graphRotatedDistanceWeight axis (angularBinCount θ) j b (u,w)) d (hdegree j b true) |
| have hf (row col : CellPath × CellPath) : F (distanceMatrixIndex row col j b true) = |
| graphRotatedDistanceWeight axis (angularBinCount θ) j b (u,w) * |
| ((U row.1 u*V row.2 w)*conj (V col.1 u*U col.2 w)) := by |
| dsimp [F,distanceMatrixIndex] |
| rw [map_mul] |
| ring |
| simpa only [hf,norm_mul,mul_pow,H,S] using he |
| change ‖∑ e ∈ H, F e‖ ≤ _ |
| rw [source_distance_graph_sum_occupied Q₁ Q₂ ρ ν A B N a n sig t hA hB H hh F,← hscale] |
| calc |
| _ ≤ ∑ j ∈ Finset.range (angularBinCount θ), ∑ b ∈ Finset.range (angularBinCount θ), |
| ‖∑ row ∈ C, ∑ col ∈ D, if distanceMatrixIndex row col j b true ∈ H |
| then F (distanceMatrixIndex row col j b true) else 0‖ := |
| (norm_sum_le _ _).trans (Finset.sum_le_sum (fun _ _ => norm_sum_le _ _)) |
| _ ≤ ∑ j ∈ Finset.range (angularBinCount θ), ∑ b ∈ Finset.range (angularBinCount θ), |
| ‖graphRotatedDistanceWeight axis (angularBinCount θ) j b (u,w)‖*S := |
| Finset.sum_le_sum (fun j _ => Finset.sum_le_sum (fun b _ => hmatrix j b)) |
| _ ≤ S := by |
| simp only [← Finset.sum_mul] |
| exact (mul_le_mul_of_nonneg_right |
| (graphRotatedDistanceWeight_norm_total_le (angularBinCount_pos hθ) axis (u,w)) hS).trans_eq (one_mul S) |
| _ = _ := by |
| dsimp [S,C,D] |
| rw [sourceProductVertices_swap_sum Q₁ Q₂ ρ ν A B n |
| (fun c p => ‖V c u‖^2*‖U p w‖^2),Finset.sum_add_distrib] |
|
|
| end PlanarFalconer |
|
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| end OAI |
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