| import OAI.MeasureTheory.Falconer.Shells.HighShellCoefficientBounds |
| import OAI.MeasureTheory.Falconer.Angular.InverseCircleArcLength |
| import OAI.MeasureTheory.Falconer.Fourier.GraphKernelBounds |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open MeasureTheory |
| open scoped ComplexConjugate |
| namespace PositiveDistanceData |
|
|
| noncomputable def inverseDescriptorList (d : PositiveDistanceData) : Finset PolicyDescriptor := |
| policyDescriptorList (fun n => d.mass n-(n:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t |
|
|
| noncomputable def inverseEndpointSymbol (d : PositiveDistanceData) (k : PairMaskIndex → ℕ) |
| (l : Fin 2) : Plane → UnitCircle → ℂ := |
| inheritedCircleSymbol d.inverseDescriptorList id (fun b => k (b,l)) d.Q d.ρ d.mass |
| d.N d.lam d.sig d.J d.i (fun _ => 1) |
|
|
| theorem inverseEndpointSymbol_measurable (d : PositiveDistanceData) (k : PairMaskIndex → ℕ) |
| (l : Fin 2) : Measurable (fun z : Plane × UnitCircle => d.inverseEndpointSymbol k l z.1 z.2) := |
| measurable_inheritedCircleSymbol _ _ _ _ _ _ _ _ _ _ _ _ measurable_const |
|
|
| theorem inverseEndpointSymbol_norm (d : PositiveDistanceData) (hQ : 0 < d.Q.side) (j : ℕ) |
| {b : ℂ × (PairMaskIndex → ℕ)} (hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)) |
| (l : Fin 2) (x : Plane) (dir : UnitCircle) : |
| ‖d.inverseEndpointSymbol b.2 l x dir‖ ≤ symbolDerivativeConstant (2*j)^(2*j) := by |
| exact inheritedCircleSymbol_norm_le _ _ _ _ hQ _ _ _ _ _ _ _ _ (2*j) |
| (pair_assignment_endpoint_budget _ _ _ (inverseLeibnizTerms_budget j _ hb) l) x (by simp) dir |
|
|
| theorem inverseCircleOperator_pair_terms (d : PositiveDistanceData) (hQ : 0 < d.Q.side) |
| (z : Plane × Plane) (j : ℕ) (dir : UnitCircle) : |
| inverseCircleOperator j (d.pairMaskSymbol z) dir = |
| ((inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)).map (fun b => |
| b.1 * ((assignmentScale d.inverseDescriptorList d.N d.sig b.2:ℂ) * |
| (d.inverseEndpointSymbol b.2 0 z.1 dir * conj (d.inverseEndpointSymbol b.2 1 z.2 dir))))).sum := by |
| have h := d.pairMaskSymbol_inverse_leibniz hQ z j (sourceAngle dir) |
| simpa only [inverseCircleOperator, sourceUnitDirection_sourceAngle, |
| inverseDescriptorList, inverseEndpointSymbol] using h |
|
|
| end PositiveDistanceData |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
