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7f37411 | 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 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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