| import OAI.MeasureTheory.Falconer.Angular.AngularCountMean |
| import OAI.MeasureTheory.Falconer.Geometry.DyadicCenterSeparation |
| import OAI.MeasureTheory.Falconer.Geometry.DyadicInverseDistance |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set Metric MeasureTheory |
|
|
| theorem parallel_inverse_distance_sum (Q : DyadicSquare) (hQ : Q.side = 1) |
| (ρ : Measure Plane) {g p : ℕ} (hgp : g ≤ p) {P : CellPath} (hP : P.length = p) : |
| ∑ c ∈ (parallelTestCells Q ρ g p P).erase P, |
| 1/dist (Q.descendant c).center (Q.descendant P).center ≤ |
| ((parallelTestCells Q ρ g p P).card:ℝ)*(2:ℝ)^g + |
| ∑ j ∈ Finset.range (p-g), |
| ((occupiedCenterCells Q ρ p (Q.descendant P).center (1/(2:ℝ)^(g+j))).card:ℝ)* |
| (2:ℝ)^(g+j+1) := by |
| classical |
| let S := (parallelTestCells Q ρ g p P).erase P |
| let d := fun c => dist (Q.descendant c).center (Q.descendant P).center |
| have hside : 0 < Q.side := by rw [hQ]; norm_num |
| have hsep (c : CellPath) (hc : c ∈ S) : 1/(2:ℝ)^p ≤ d c := by |
| have hc' := Finset.mem_of_mem_erase hc |
| have hl := (mem_dyadicPaths p c).mp (Finset.mem_filter.mp hc').1 |
| simpa only [hQ] using dyadic_centers_separated Q hside hl hP (Finset.ne_of_mem_erase hc) |
| have hd (c : CellPath) (hc : c ∈ S) : 0 < d c := |
| lt_of_lt_of_le (by positivity : 0 < 1/(2:ℝ)^p) (hsep c hc) |
| have hh := sum_inverse_distance_dyadic S d g (p-g) hd (fun c hc => by |
| simpa only [Nat.add_sub_of_le hgp] using hsep c hc) |
| apply hh.trans |
| apply add_le_add |
| · apply mul_le_mul_of_nonneg_right _ (by positivity) |
| exact_mod_cast Finset.card_le_card (Finset.erase_subset P (parallelTestCells Q ρ g p P)) |
| · apply Finset.sum_le_sum |
| intro j _ |
| apply mul_le_mul_of_nonneg_right _ (by positivity) |
| apply Nat.cast_le.mpr |
| apply Finset.card_le_card |
| intro c hc |
| obtain ⟨hc,hdc⟩ := Finset.mem_filter.mp hc |
| have hc' := Finset.mem_filter.mp (Finset.mem_of_mem_erase hc) |
| exact Finset.mem_filter.mpr ⟨hc'.1,hc'.2.1,hdc⟩ |
|
|
| theorem parallel_inverse_distance_profile (Q : DyadicSquare) (hQ : Q.side = 1) |
| (ρ : Measure Plane) (m : ℝ → ℝ) (N : ℕ) (lam : ℝ) |
| (hpack : ∀ g p : ℕ, g ≤ p → p ≤ N → ∀ x : Plane, ∀ C₀ : ℝ, 0 ≤ C₀ → |
| ((occupiedCenterCells Q ρ p x (C₀/(2:ℝ)^g)).card:ℝ) ≤ |
| (4*(C₀+1)^2)*((2:ℝ)^N)^(m p-m g+6*lam)) |
| {g p : ℕ} (hgp : g ≤ p) (hpN : p ≤ N) {P : CellPath} (hP : P.length = p) : |
| ∑ c ∈ (parallelTestCells Q ρ g p P).erase P, |
| 1/dist (Q.descendant c).center (Q.descendant P).center ≤ |
| 484*(((2:ℝ)^N)^(m p-m g+6*lam)*(2:ℝ)^g + |
| ∑ j ∈ Finset.range (p-g), |
| ((2:ℝ)^N)^(m p-m (g+j)+6*lam)*(2:ℝ)^(g+j+1)) := by |
| classical |
| apply (parallel_inverse_distance_sum Q hQ ρ hgp hP).trans |
| rw [mul_add,Finset.mul_sum] |
| apply add_le_add |
| · have hh := hpack g p hgp hpN (Q.descendant P).center 10 (by norm_num) |
| change ((parallelTestCells Q ρ g p P).card:ℝ) ≤ _ at hh |
| norm_num at hh |
| simpa only [mul_assoc] using |
| mul_le_mul_of_nonneg_right hh (by positivity : 0 ≤ (2:ℝ)^g) |
| · apply Finset.sum_le_sum |
| intro j hj |
| have hj' := Finset.mem_range.mp hj |
| have hh := hpack (g+j) p (by omega) hpN (Q.descendant P).center 1 (by norm_num) |
| norm_num at hh |
| have hpos : 0 ≤ ((2:ℝ)^N)^(m p-m (g+j)+6*lam) := by positivity |
| have hh' : ((occupiedCenterCells Q ρ p (Q.descendant P).center (1/(2:ℝ)^(g+j))).card:ℝ) ≤ |
| 484*((2:ℝ)^N)^(m p-m (g+j)+6*lam) := by |
| simp only [one_div] |
| linarith |
| simpa only [mul_assoc] using |
| mul_le_mul_of_nonneg_right hh' (by positivity : 0 ≤ (2:ℝ)^(g+j+1)) |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
