| import OAI.MeasureTheory.Falconer.Estimates.SuffixScales |
| import OAI.MeasureTheory.Falconer.Graphs.EligibleDyadicGraph |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace Falconer |
| open Set MeasureTheory Metric |
| open scoped ENNReal |
| attribute [local instance] Classical.propDecidable |
| variable {d : ℕ} |
|
|
| theorem parallelGoodSet_local_test (Q : DyadicCube d) (hQ : Q.side = 1) |
| (ρ : Measure (Space d)) {a g p : ℕ} (hag : a ≤ g) (hgp : g ≤ p) |
| (X P : List (Fin d → Bool)) (hX : X.length = a) |
| (w : Sphere d) (W T : ℝ) (hT : 0 ≤ T) |
| (hw : w ∈ Regularization.parallelGoodSet Q ρ g p (X++P) W T) : |
| DyadicParallelTest (Q.descendant X) ((Q.descendant X).occupiedPaths ρ (p-a)) |
| (g-a) P w W ⌈T⌉₊ := by |
| have hscale : (Q.descendant X).side/(2:ℝ)^(g-a) = 1/(2:ℝ)^g := by |
| rw [Q.descendant_side,hX,hQ,div_div,← pow_add,Nat.add_sub_of_le hag] |
| let S := ((Q.descendant X).occupiedPaths ρ (p-a)).filter fun z => |
| z.take (g-a) = P.take (g-a) ∧ |
| ‖((Q.descendant X).descendant z).center-((Q.descendant X).descendant P).center‖ ≤ |
| Real.sqrt d*((Q.descendant X).side/(2:ℝ)^(g-a)) ∧ |
| ‖perpendicularPart (w:Space d) |
| (((Q.descendant X).descendant z).center-((Q.descendant X).descendant P).center)‖ ≤ W |
| have hinj : Function.Injective (fun z : List (Fin d → Bool) => X++z) := by |
| intro u v huv |
| exact List.append_cancel_left huv |
| have hsub : S.image (fun z => X++z) ⊆ |
| (Regularization.parallelTestCells Q ρ g p (X++P)).filter (fun z => |
| ‖perpendicularPart (w:Space d) ((Q.descendant z).center-(Q.descendant (X++P)).center)‖ ≤ W) := by |
| intro z hz |
| obtain ⟨u,hu,rfl⟩ := Finset.mem_image.mp hz |
| obtain ⟨hu,_,hdist,hperp⟩ := Finset.mem_filter.mp hu |
| obtain ⟨hu,hocc⟩ := Finset.mem_filter.mp hu |
| have hulen := (mem_cubePaths _ _ _).mp hu |
| apply Finset.mem_filter.mpr |
| constructor |
| · simp only [Regularization.parallelTestCells,Regularization.occupiedCenterCells, |
| Finset.mem_filter] |
| refine ⟨?_,?_,?_⟩ |
| · apply (mem_cubePaths _ _ _).mpr |
| simp only [List.length_append,hX,hulen] |
| omega |
| · simpa only [Q.descendant_append] using hocc |
| · rw [mem_ball,dist_eq_norm,Q.descendant_append,Q.descendant_append] |
| apply hdist.trans_lt |
| rw [hscale] |
| have hs := sqrt_dimension_le d |
| have hr : Real.sqrt d < 10*((d:ℝ)+1) := by |
| nlinarith [Nat.cast_nonneg (α := ℝ) d] |
| simpa only [div_eq_mul_inv,one_mul] using |
| mul_lt_mul_of_pos_right hr (by positivity : 0 < 1/(2:ℝ)^g) |
| · simpa only [Q.descendant_append] using hperp |
| have hcard : S.card ≤ Regularization.parallelTestCount Q ρ g p (X++P) W w := by |
| calc |
| S.card = (S.image (fun z => X++z)).card := |
| (Finset.card_image_of_injective S hinj).symm |
| _ ≤ _ := Finset.card_le_card hsub |
| have hcount : (Regularization.parallelTestCount Q ρ g p (X++P) W w : ℝ) ≤ T := by |
| unfold Regularization.parallelGoodSet at hw |
| split_ifs at hw with hocc |
| · simpa only [ENNReal.toReal_natCast,ENNReal.toReal_ofReal hT] using |
| ENNReal.toReal_mono ENNReal.ofReal_ne_top hw |
| · exact False.elim hw |
| exact hcard.trans ((Nat.cast_le (α := ℝ)).mp (hcount.trans (Nat.le_ceil T))) |
|
|
| theorem descriptorEndpointGood_graph_test (Q : DyadicCube d) (hQ : Q.side = 1) |
| (ρ : Measure (Space d)) (m : ℝ → ℝ) (N : ℕ) (σ : ℝ) (J i : ℕ) |
| {a g p : ℕ} (hag : a ≤ g) (hgp : g ≤ p) |
| (X P : List (Fin d → Bool)) (hX : X.length = a) (hP : P.length = p-a) |
| {x : Space d} (hx : x ∈ ((Q.descendant X).descendant P).carrier) |
| (w : Sphere d) |
| (hw : (x,w) ∈ descriptorEndpointGood Q ρ m N σ J i (.parallel g p)) : |
| DyadicParallelTest (Q.descendant X) ((Q.descendant X).occupiedPaths ρ (p-a)) |
| (g-a) P w (angularTestWidth N σ J i p) |
| ⌈((2:ℝ)^N)^(PlanarFalconer.height |
| (fun n : ℕ => m n-((d:ℝ)/2)*(n:ℝ)/N) g p+1000*((d:ℝ)+1)*σ)⌉₊ := by |
| apply parallelGoodSet_local_test Q hQ ρ hag hgp X P hX w _ _ (by positivity) |
| apply (descriptorEndpointGood_on_cell Q ρ m N σ J i (.parallel g p) (X++P) ?_ |
| (by simpa only [Q.descendant_append] using hx) w).mp hw |
| simp only [PlanarFalconer.descriptorAnchor,List.length_append,hX,hP] |
| omega |
|
|
| theorem descriptorMask_graph_test (Q : DyadicCube d) (hQ : Q.side = 1) |
| (ρ : Measure (Space d)) (m : ℝ → ℝ) (N : ℕ) (σ : ℝ) (J i : ℕ) |
| {a g p : ℕ} (hag : a ≤ g) (hgp : g ≤ p) |
| (X P : List (Fin d → Bool)) (hX : X.length = a) (hP : P.length = p-a) |
| (S : Finset (Sphere d)) |
| (hcover : ∀ w : Sphere d, ∃ c ∈ S, |
| dist w c ≤ descriptorMaskRadius N σ (.parallel g p)/2) |
| {x : Space d} (hx : x ∈ ((Q.descendant X).descendant P).carrier) (w : Sphere d) |
| (hw : descriptorMask Q ρ m N σ J i (.parallel g p) S x w ≠ 0) : |
| DyadicParallelTest (Q.descendant X) ((Q.descendant X).occupiedPaths ρ (p-a)) |
| (g-a) P w (angularTestWidth N σ J (i+1) p) |
| ⌈((2:ℝ)^N)^(PlanarFalconer.height |
| (fun n : ℕ => m n-((d:ℝ)/2)*(n:ℝ)/N) g p+1000*((d:ℝ)+1)*σ)⌉₊ := |
| descriptorEndpointGood_graph_test Q hQ ρ m N σ J (i+1) hag hgp X P hX hP hx w |
| (descriptorMask_support Q ρ m N σ J i (.parallel g p) S hcover w hw) |
|
|
| end Falconer |
|
|
| end OAI |
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