| import OAI.MeasureTheory.Falconer.Estimates.FarCellCount |
| import OAI.MeasureTheory.Falconer.Shells.PositiveSpatialKernel |
|
|
| namespace OAI |
|
|
| /-! # Positive far kernels over the actual comparable cell pairs |
|
|
| The maximum ranges over the original tested distance laws. The number |
| of summands is bounded by two free cell labels, not four. |
| -/ |
|
|
| namespace Falconer |
| open Set MeasureTheory |
| open scoped ENNReal |
| attribute [local instance] Classical.propDecidable |
|
|
| noncomputable def farCollisionMaximum {d : ℕ} |
| (w₀ : Sphere d) (Q : DyadicCube d) (ρ : Measure (Space d)) |
| (m : ℝ → ℝ) (A : ℕ → ℝ) (N : ℕ) (σ : ℝ) (J i q b t : ℕ) |
| (X Y : List (Fin d → Bool)) (n : ℕ) : ℝ≥0∞ := |
| ((farCellPartners (Q.descendant X) ρ n) ×ˢ |
| (farCellPartners (Q.descendant Y) ρ n)).sup fun z => |
| max (positiveCollisionCount w₀ Q ρ m A N σ J i q b t (X++z.1.1) (X++z.1.2)) |
| (positiveCollisionCount w₀ Q ρ m A N σ J i q b t (Y++z.2.1) (Y++z.2.2)) |
|
|
| theorem far_positive_kernel_sum_le {d : ℕ} |
| (w₀ : Sphere d) (Q : DyadicCube d) (ρ : Measure (Space d)) [IsFiniteMeasure ρ] |
| (m : ℝ → ℝ) (A : ℕ → ℝ) (N : ℕ) (σ : ℝ) (J i q b t : ℕ) |
| (X Y : List (Fin d → Bool)) (n : ℕ) |
| (S : (List (Fin d → Bool) × List (Fin d → Bool)) → |
| (List (Fin d → Bool) × List (Fin d → Bool)) → |
| Set ((Space d × Space d) × (Space d × Space d))) : |
| (∑ z ∈ (farCellPartners (Q.descendant X) ρ n) ×ˢ |
| (farCellPartners (Q.descendant Y) ρ n), |
| ENNReal.ofReal ((2:ℝ)^(t-b)*((2:ℝ)^N)^(4*m b)) * |
| (∫⁻ v in S z.1 z.2, |
| ENNReal.ofReal (1/(1+(2:ℝ)^t*|dist v.1.1 v.1.2-dist v.2.1 v.2.2|)^2) |
| ∂(testedCellPair ρ ρ (Q.descendant (X++z.1.1)) (Q.descendant (X++z.1.2)) |
| (policyTestPairs w₀ Q ρ m A N σ J i q b t)).prod |
| (testedCellPair ρ ρ (Q.descendant (Y++z.2.1)) (Q.descendant (Y++z.2.2)) |
| (policyTestPairs w₀ Q ρ m A N σ J i q b t)))) ≤ |
| (((farCellPartners (Q.descendant X) ρ n) ×ˢ |
| (farCellPartners (Q.descendant Y) ρ n)).card : ℝ≥0∞) * |
| PlanarFalconer.binKernelConstant * |
| farCollisionMaximum w₀ Q ρ m A N σ J i q b t X Y n := by |
| let T := (farCellPartners (Q.descendant X) ρ n) ×ˢ |
| (farCellPartners (Q.descendant Y) ρ n) |
| calc |
| _ ≤ ∑ _z ∈ T, PlanarFalconer.binKernelConstant * |
| farCollisionMaximum w₀ Q ρ m A N σ J i q b t X Y n := by |
| apply Finset.sum_le_sum |
| intro z hz |
| exact (positiveCollision_spatial_kernel_max w₀ Q ρ m A N σ J i q b t |
| (X++z.1.1) (X++z.1.2) (Y++z.2.1) (Y++z.2.2) (S z.1 z.2)).trans |
| (mul_le_mul' le_rfl (Finset.le_sup (s := T) (f := fun z => |
| max (positiveCollisionCount w₀ Q ρ m A N σ J i q b t (X++z.1.1) (X++z.1.2)) |
| (positiveCollisionCount w₀ Q ρ m A N σ J i q b t (Y++z.2.1) (Y++z.2.2))) hz)) |
| _ = _ := by simp only [Finset.sum_const, nsmul_eq_mul, mul_assoc, T] |
|
|
| theorem farCellPartners_regular_card {d : ℕ} {Q : DyadicCube d} |
| {ρ : Measure (Space d)} {N : ℕ} {s lam : ℝ} {m : ℝ → ℝ} |
| (h : Regularization.RegularizedProfileData Q ρ N s lam m) (hQ : 0 < Q.side) |
| {a b : ℕ} (hab : a ≤ b) (hb : b ≤ N) |
| (X Y : List (Fin d → Bool)) (hX : X.length = a) (hY : Y.length = a) : |
| (((farCellPartners (Q.descendant X) ρ (b-a)) ×ˢ |
| (farCellPartners (Q.descendant Y) ρ (b-a))).card : ℝ) ≤ |
| ((2*(1100*(d+1))+1)^(2*d):ℕ) * |
| ((2:ℝ)^N)^(2*(m ((b:ℝ)/N)-m ((a:ℝ)/N))+4*((d:ℝ)+1)*lam) := by |
| have hc := farCellPartners_pair_card (Q.descendant X) (Q.descendant Y) |
| (Q.descendant_side_pos hQ X) (Q.descendant_side_pos hQ Y) ρ ρ (b-a) |
| have hc' : (((farCellPartners (Q.descendant X) ρ (b-a)) ×ˢ |
| (farCellPartners (Q.descendant Y) ρ (b-a))).card : ℝ) ≤ |
| (((2*(1100*(d+1))+1)^(2*d):ℕ):ℝ) * |
| (((Q.descendant X).occupiedPaths ρ (b-a) ×ˢ |
| (Q.descendant Y).occupiedPaths ρ (b-a)).card : ℝ) := by exact_mod_cast hc |
| exact hc'.trans (mul_le_mul_of_nonneg_left (h.pair_count hab hb X Y hX hY) |
| (Nat.cast_nonneg _)) |
|
|
| end Falconer |
|
|
| end OAI |
|
|
