| import OAI.MeasureTheory.Falconer.Shells.HighShellFourierError |
| import OAI.MeasureTheory.Falconer.Estimates.ProfileMassError |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| namespace PositiveDistanceData |
|
|
| theorem normalized_cellPair_mass (d : PositiveDistanceData) [IsFiniteMeasure d.ρ] : |
| d.levelMass^(-2:ℤ)*d.cellPair.real univ = |
| ((d.ρ (d.Q.descendant d.X).carrier).toReal/d.levelMass)* |
| ((d.ρ (d.Q.descendant d.Y).carrier).toReal/d.levelMass) := by |
| have hp : d.cellPair univ = |
| d.ρ (d.Q.descendant d.X).carrier*d.ρ (d.Q.descendant d.Y).carrier := by |
| unfold cellPair |
| rw [← univ_prod_univ,Measure.prod_prod,Measure.restrict_apply_univ, |
| Measure.restrict_apply_univ] |
| rw [measureReal_def,hp,ENNReal.toReal_mul] |
| simp only [zpow_neg,zpow_ofNat,div_eq_mul_inv] |
| ring |
|
|
| theorem normalized_cellPair_profile_mass (d : PositiveDistanceData) {B : ℕ} {δ : ℝ} |
| (hd : RegularizedProfileData d.Q d.ρ d.N B δ d.lam d.mass) |
| (ha : d.a ≤ d.N) (hX : d.X.length = d.a) (hY : d.Y.length = d.a) : |
| d.levelMass^(-2:ℤ)*d.cellPair.real univ ≤ ((2:ℝ)^d.N)^(6*d.lam) := by |
| let := hd.probability |
| rw [d.normalized_cellPair_mass] |
| have h := mul_le_mul (hd.normalized_cell_mass_le ha hX) |
| (hd.normalized_cell_mass_le ha hY) (by positivity) (by positivity) |
| rw [← Real.rpow_add (by positivity : 0 < (2:ℝ)^d.N)] at h |
| convert h using 1 |
| congr 1 |
| ring_nf |
|
|
| theorem uniform_high_shell_profile_error {sig lam T : ℝ} (hsig : 0 < sig) : |
| ∃ 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 : ℝ, (d.t:ℝ)-d.sig*d.N ≤ v → (2:ℝ)^(v-1) ≤ r → |
| ‖charFun d.weightedDistance r-d.normalizedHighShellF n v r‖ ≤ ((2:ℝ)^d.N)^(-T) := by |
| obtain ⟨n,N₀,h₀⟩ := uniform_high_shell_fourier_error (T := T+6*lam) hsig |
| refine ⟨n,N₀,?_⟩ |
| intro d B δ hd hN hsig' hlam hQ hs hq hqs hw ht hX hY v r hv hr |
| let := hd.probability |
| have he := h₀ d hN hsig' hQ hs hq hqs hw ht v r hv hr |
| have hm := d.normalized_cellPair_profile_mass hd (by omega) hX hY |
| rw [hlam] at hm |
| apply he.trans |
| apply (mul_le_mul_of_nonneg_right hm (by positivity : 0 ≤ ((2:ℝ)^d.N)^(-(T+6*lam)))).trans_eq |
| rw [← Real.rpow_add (by positivity : 0 < (2:ℝ)^d.N)] |
| congr 1 |
| ring |
|
|
| end PositiveDistanceData |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
