| import Mathlib |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.FalconerCompletion.Angular.InnerNormalizedOfClose |
| import OAI.MeasureTheory.FalconerCompletion.Fourier.MinimalMask |
|
|
| namespace OAI |
|
|
| /-! Dyadic shells and prefix decompositions. -/ |
|
|
| open _root_.OAI.PlanarFalconer |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace CurrentFalconer.TreeND |
| open Set MeasureTheory PlanarFalconer _root_.OAI.GraphLiteralSupport.Attempt4FullFar |
| open scoped ContDiff Topology |
| variable {d : ℕ} |
| namespace PositiveDistanceState |
| variable (e : PositiveDistanceState d) |
|
|
| noncomputable def prefixCutoff : EuclideanSpace ℝ (Fin d) → ℂ := |
| separatedCutoff e.a (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ)) |
|
|
| lemma prefixCutoff_contDiff : ContDiff ℝ ∞ e.prefixCutoff := separatedCutoff_contDiff _ _ _ |
| lemma prefixCutoff_norm_le (w : EuclideanSpace ℝ (Fin d)) : ‖e.prefixCutoff w‖≤1 := |
| separatedCutoff_norm_le _ _ _ w |
|
|
| lemma ae_prefixCutoff_local [SFinite e.ρ] {K : ℝ} (hK : 32*(d+1:ℝ)≤K) |
| (hX : e.X.length=e.a) (hY : e.Y.length=e.a) |
| (hs : farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ))) : |
| ∀ᵐ z ∂e.cellPair, |
| e.prefixCutoff =ᶠ[𝓝 ((dist z.1 z.2)⁻¹ • (z.1-z.2))] (fun _ => 1) ∧ |
| e.prefixCutoff =ᶠ[𝓝 (-((dist z.1 z.2)⁻¹ • (z.1-z.2)))] (fun _ => 0) := by |
| filter_upwards [e.ae_cellPair_mem] with z hz |
| have hx : z.1∈Attempt4FullFar.ambientGridCell e.a (fun j => (pathIndex e.X j:ℤ)) := by |
| have hx := hz.1 |
| change z.1∈Attempt4FullFar.ambientGridCell e.X.length (fun j => (pathIndex e.X j:ℤ)) at hx |
| rwa [hX] at hx |
| have hy : z.2∈Attempt4FullFar.ambientGridCell e.a (fun j => (pathIndex e.Y j:ℤ)) := by |
| have hy := hz.2 |
| change z.2∈Attempt4FullFar.ambientGridCell e.Y.length (fun j => (pathIndex e.Y j:ℤ)) at hy |
| rwa [hY] at hy |
| simpa only [prefixCutoff,radialVector,dist_eq_norm] using separatedCutoff_local hK hs hx hy |
|
|
| end PositiveDistanceState |
|
|
| lemma uniform_prefixCutoff_derivatives (K : ℝ) : |
| ∃ C : ℕ → ℝ, (∀ m, 0≤C m) ∧ ∀ (e : PositiveDistanceState d), |
| farSeparatedGridND e.a K (fun j => (pathIndex e.X j:ℤ)) (fun j => (pathIndex e.Y j:ℤ)) → |
| ∀ m w, ‖w‖=1 → ‖iteratedFDeriv ℝ m e.prefixCutoff w‖≤C m := by |
| obtain ⟨C,hC,h⟩ := uniform_separatedCutoff_derivatives (d:=d) K |
| exact ⟨C,hC,fun e hs m w hw => h _ _ _ hs m w hw⟩ |
|
|
| end CurrentFalconer.TreeND |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
|
|
