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b224933 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | 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
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