| import OAI.MeasureTheory.Falconer.Shells.HighShellEquation21 |
| import OAI.MeasureTheory.Falconer.Shells.HighShellSupremum |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set MeasureTheory |
| namespace PositiveDistanceData |
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| theorem uniform_high_shell_equation21 {sig lam eps T : ℝ} |
| (hsig : 0 < sig) (heps : 0 < eps) : |
| ∃ n N₀ : ℕ, ∀ d : PositiveDistanceData, ∀ (B : ℕ) (δ : ℝ), |
| RegularizedProfileData d.Q d.ρ d.N B δ d.lam d.mass → |
| N₀ ≤ d.N → d.sig = sig → d.lam = lam → |
| 0 < d.Q.side → d.StronglySeparated → 0 < d.q → |
| d.sig*d.N ≤ (d.q:ℝ) → d.a+100*d.q < d.t → d.t ≤ d.N → |
| d.X.length = d.a → d.Y.length = d.a → |
| ∀ v : ℝ, (d.t:ℝ)-d.sig*d.N ≤ v → v ≤ d.t → |
| ∀ Ψ : ℝ → ℝ, Measurable Ψ → (∀ x, 0 ≤ Ψ x) → |
| (∀ x, Ψ x ≠ 0 → x ∈ Ioo (1/4:ℝ) 4) → |
| ∀ W : ℝ, 0 ≤ W → (∀ x, Ψ x ≤ W) → |
| (∀ x ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ x) → |
| (2:ℝ)^(-(d.a:ℝ)) * |
| (∫ r in Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)), ‖charFun d.weightedDistance r‖^2) ≤ |
| ((2:ℝ)^d.N)^eps*d.inverseShellSupremum n v Ψ+((2:ℝ)^d.N)^(-T) := by |
| obtain ⟨n,N₀,h₀⟩ := uniform_high_shell_parent_bound (lam := lam) (T := T) hsig heps |
| refine ⟨n,N₀,?_⟩ |
| intro d B δ hd hN hdSig hdLam hQ hsep hq hqs hw ht hX hY v hv hvt Ψ hΨ hpos hs W hW hΨW hone |
| let := hd.probability |
| have hsupport : ∀ x, 4 < x → Ψ x = 0 := by |
| intro x hx |
| by_contra hn |
| exact (not_lt_of_ge hx.le) (hs x hn).2 |
| have hsup := d.inverseShellSupremum_majorizes hQ n v Ψ hW hΨW hsupport |
| exact h₀ d B δ hd hN hdSig hdLam hQ hsep hq hqs hw ht hX hY v hv hvt |
| Ψ hΨ hpos hs W hW hΨW hone _ hsup.1 hsup.2 |
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| end PositiveDistanceData |
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| end PlanarFalconer |
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| end OAI |
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