| import OAI.MeasureTheory.Falconer.Recursion.DescriptorMasks |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
|
|
| theorem uniform_descriptor_good_transfer {sig : ℝ} (hsig : 0 < sig) |
| {J : ℕ} (hJ : 0 < J) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (Q : DyadicSquare), Q.side = 1 → |
| ∀ (ρ : Measure Plane) (m : ℝ → ℝ) (lam : ℝ) (i : ℕ) |
| (b : PolicyDescriptor), ValidDescriptor N b → |
| ∀ P : CellPath, P.length = descriptorAnchor b → ∀ ω ω' : UnitCircle, |
| dist ω ω' ≤ sourceAngularRadius N sig (descriptorLength b) → |
| ω ∈ descriptorGoodSet Q ρ m N lam sig J i P b → |
| ω' ∈ descriptorGoodSet Q ρ m N lam sig J (i+1) P b := by |
| obtain ⟨N₁,h₁⟩ := uniform_parallel_width_transfer hsig hJ (C₀ := 1) zero_le_one |
| obtain ⟨N₂,h₂⟩ := uniform_projection_width_transfer hsig hJ (C₀ := 1) zero_le_one |
| refine ⟨max N₁ N₂,?_⟩ |
| intro N hN Q hQ ρ m lam i b hb P hP ω ω' hd hg |
| obtain ⟨hN₁, hN₂⟩ := max_le_iff.mp hN |
| change dist (ω:Plane) (ω':Plane) ≤ _ at hd |
| cases b with |
| | parallel g p => |
| dsimp only [descriptorGoodSet,parallelGoodSet,Set.mem_ofPred_eq] at hg ⊢ |
| change (_ : ℝ) ≤ _ at hg ⊢ |
| apply le_trans (Nat.cast_le.mpr (h₁ N hN₁ |
| Q ρ g p hb.1 P i ω ω' (by simpa only [descriptorLength,one_mul] using hd))) hg |
| | projection p u => |
| dsimp only [descriptorGoodSet,projectionGoodSet,Set.mem_ofPred_eq] at hg ⊢ |
| change (_ : ℝ) ≤ _ at hg ⊢ |
| apply le_trans (Nat.cast_le.mpr (h₂ N hN₂ |
| Q hQ ρ p u hb.1 P hP i ω ω' (by simpa only [descriptorLength,one_mul] using hd))) hg |
|
|
| end PlanarFalconer |
|
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| end OAI |
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|
