| import Mathlib |
| import OAI.Geometry.Cannon.ScalarLimits.InteriorScalar |
|
|
| namespace OAI |
|
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| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
| open Set Metric Topology |
| open Set Topology unitInterval |
| open scoped BigOperators |
| open Set unitInterval |
| open Set MeasureTheory |
| open Set Filter Metric Topology |
| open Set Filter Topology |
| open Set Filter MeasureTheory Topology |
| open scoped NNReal ENNReal |
|
|
| namespace CannonApprox |
|
|
| open CannonGrid CannonChart CannonCharts CannonScales CannonPacking CannonAnnular |
| open Set Metric Filter Topology |
|
|
| theorem geometric_annular_compact_tails {X : Type*} [MetricSpace X] [CompactSpace X] |
| [LocallyPathConnectedSpace X] |
| (φ : X ≃ₜ SphereTwo) (d : ℝ) (hd : 0 < d) (hdia : d < diam (univ : Set X)) |
| (a : ℝ) (ha : 1 < a) (Q : ℕ) (CP C c L : ℝ) |
| (hpack : PolynomialPacking (X := X) Q CP) (hCP : 0 ≤ CP) |
| (hC : 0 ≤ C) (hc : 0 < c) (hL : 1 ≤ L) (hdc : d ≤ c) |
| (hpaths : ∀ (o u v : X) (s : ℝ), 0 < s → dist o u ≤ s → dist o v ≤ s → |
| JoinedIn {z | dist o z < L*s} u v) |
| (hexp : ∀ (o : X) (l : ℕ), ∃ h : X → X, Continuous h ∧ |
| ∀ u v, dist u o ≤ 3*(a^l)⁻¹ → dist v o ≤ 3*(a^l)⁻¹ → |
| dist (h u) (h v) ≤ C/(a^l)⁻¹*dist u v ∧ |
| c/(a^l)⁻¹*dist u v ≤ dist (h u) (h v)) |
| (ι : ℕ → Type*) [∀ n, Fintype (ι n)] (p : ∀ n, ι n → X) |
| (hpsep : ∀ n i j, i ≠ j → (a^n)⁻¹ ≤ dist (p n i) (p n j)) |
| (hpnet : ∀ n x, ∃ i, dist (p n i) x ≤ (a^n)⁻¹) |
| (e : Square → X) (he : IsEmbedding e) (N : ℕ → ℕ) |
| (ρ : ∀ n, ι (N n) → ℝ) (hρ : ∀ n i, 0 ≤ ρ n i) |
| (hadm : ∀ n, Admissible (fun i => e ⁻¹' ball (p (N n) i) (2*(a^(N n))⁻¹)) (ρ n)) |
| (m : ℕ → ℕ) (hm : ∀ n, 0 < m n) |
| (hmesh : ∀ n z, ∀ i : Index (3*m n), weight (3*m n) z i ≠ 0 → |
| dist (e z) (e (vertex (3*m n) (by have := hm n; omega) i)) < (a^(N n))⁻¹) : |
| let M := fun n => CannonCells.modulus (fun i => ball (p n i) (2*(a^n)⁻¹)) (macroFamily d) |
| let f := fun n => interiorScalar e he (p (N n)) ((a^(N n))⁻¹) |
| (ρ n) (hρ n) (hadm n) (m n) (hm n) |
| ∃ E : ℝ, 0 ≤ E ∧ ∀ K : Set X, IsCompact K → K ⊆ interior (range e) → |
| ∃ l₀ m₀ : ℕ, ∀ l, l₀ ≤ l → ∀ n, l+m₀ ≤ N n → ∀ o ∈ K, |
| ∀ x y : interior (range e), |
| dist x.val o ≤ (a^l)⁻¹/(2*L) → dist y.val o ≤ (a^l)⁻¹/(2*L) → |
| (f n x-f n y)^2 ≤ E*M (N n-l)* |
| (∑ i ∈ Finset.univ.filter (fun i => |
| (a^l)⁻¹/4 ≤ dist (p (N n) i) o ∧ dist (p (N n) i) o ≤ 4*(a^l)⁻¹), |
| (ρ n i)^2) := by |
| classical |
| let M := fun n => CannonCells.modulus (fun i => ball (p n i) (2*(a^n)⁻¹)) (macroFamily d) |
| let f := fun n => interiorScalar e he (p (N n)) ((a^(N n))⁻¹) |
| (ρ n) (hρ n) (hadm n) (m n) (hm n) |
| obtain ⟨E,hE,hbound⟩ := exists_scalar_annular_constant Q CP C c L hpack hCP hC hc hL hpaths |
| obtain ⟨d₁,hd₁,hdiam⟩ := exists_chart_lateral_diameter e he |
| obtain ⟨m₀,hm₀⟩ := exists_ratio_threshold ha (L*(L*2+1)+3) (max (C+2) (max c 1)) |
| have hap : 0 < a := zero_lt_one.trans ha |
| have hLp : 0 < L := zero_lt_one.trans_le hL |
| have hcover (k : ℕ) (x : X) : ∃ i, x ∈ ball (p k i) (2*(a^k)⁻¹) := by |
| obtain ⟨i,hi⟩ := hpnet k x |
| refine ⟨i,?_⟩ |
| rw [mem_ball,dist_comm] |
| exact hi.trans_lt (by have := inv_pos.mpr (pow_pos hap k); linarith) |
| refine ⟨E,hE,fun K hK hKU => ?_⟩ |
| obtain ⟨δ,hδ,hδU⟩ := hK.exists_thickening_subset_open isOpen_interior hKU |
| have hscl := inverse_geometric_tendsto_zero ha |
| have hsmall : ∀ᶠ l in atTop, 4*(a^l)⁻¹ < min δ d₁ := |
| (tendsto_const_nhds.mul hscl).eventually (gt_mem_nhds (by simpa using lt_min hδ hd₁)) |
| obtain ⟨l₀,hl₀⟩ := eventually_atTop.mp hsmall |
| refine ⟨l₀,m₀,fun l hl n hn o ho x y hx hy => ?_⟩ |
| let R := (a^l)⁻¹ |
| let r := (a^(N n))⁻¹ |
| have hR : 0 < R := inv_pos.mpr (pow_pos hap l) |
| have hr : 0 < r := inv_pos.mpr (pow_pos hap (N n)) |
| have hln : l ≤ N n := by omega |
| have hrat : r/R = (a^(N n-l))⁻¹ := inverse_scale_ratio hap hln |
| have hconstraints := hm₀ (N n-l) (by omega) |
| rw [← hrat] at hconstraints |
| have hRone : R ≤ 1 := inv_le_one_of_one_le₀ (one_le_pow₀ ha.le) |
| have hRd : 4*R < d₁ := (hl₀ l hl).trans_le (min_le_right _ _) |
| have hchart : ball o (4*R) ⊆ range e := by |
| intro z hz |
| apply interior_subset |
| apply hδU |
| exact mem_thickening_iff.mpr ⟨o,ho, |
| (mem_ball.mp hz).trans ((hl₀ l hl).trans_le (min_le_left _ _))⟩ |
| have hrR : r ≤ R/2 := by |
| have hh := (div_le_iff₀ hR).mp hconstraints.1 |
| linarith |
| have hAr : (L*(L*2+1)+3)*r ≤ R/4 := by |
| have hh := (div_le_iff₀ hR).mp (show ((L*(L*2+1)+3)*r)/R ≤ 1/4 by |
| simpa only [mul_div_assoc] using hconstraints.2.1) |
| linarith |
| obtain ⟨h,hh,hlocal⟩ := hexp o l |
| obtain ⟨_,ω,hω,hωE⟩ := sphere_macro_modulus_minimum φ d hd hdia (p (N n-l)) |
| ((a^(N n-l))⁻¹) (hcover (N n-l)) |
| have hω' : MacroscopicAdmissible (p (N n-l)) (r/R) d ω := by |
| simpa only [hrat] using hω |
| obtain ⟨zx,hzx⟩ := interior_subset x.property |
| obtain ⟨zy,hzy⟩ := interior_subset y.property |
| have hzdist : dist o (e zy) ≤ R/(2*L) := by |
| rw [hzy,dist_comm] |
| exact hy |
| have hzero : dist o (e zx) ≤ R/(2*L) := by |
| rw [hzx,dist_comm] |
| exact hx |
| have hlarge (u v : X) (hu : dist u o ≤ 2*R) (hv : dist v o ≤ 2*R) |
| (huv : R ≤ dist u v) : d ≤ dist (h u) (h v) := by |
| have hu' : dist u o ≤ 3*R := by linarith |
| have hv' : dist v o ≤ 3*R := by linarith |
| have hlow := (hlocal u v hu' hv').2 |
| calc |
| d ≤ c := hdc |
| _ = c/R*R := by field_simp |
| _ ≤ c/R*dist u v := mul_le_mul_of_nonneg_left huv (div_nonneg hc.le hR.le) |
| _ ≤ dist (h u) (h v) := hlow |
| have hs : 0 < R/(2*L) := by positivity |
| have hLs : L*(R/(2*L)) ≤ R/2 := by field_simp; linarith |
| have hmain := hbound e he d₁ hdiam (p (N n-l)) (p (N n)) h hh o R r d (R/(2*L)) |
| hR hRone hRd hr hrR hAr hconstraints.2.2 hs hLs hchart |
| (fun i j hij => by rw [hrat]; exact hpsep (N n-l) i j hij) |
| (hpsep (N n)) (hpnet (N n)) |
| (fun u v hu hv => (hlocal u v hu hv).1) |
| (fun u v hu hv => (hlocal u v hu hv).2) hlarge ω hω' (ρ n) (hρ n) |
| (hadm n) (m n) (hm n) (hmesh n) zy zx hzdist hzero |
| rw [hωE] at hmain |
| have hfx : f n x = scalar (fun i => e ⁻¹' ball (p (N n) i) (2*r)) (ρ n) (m n) (hm n) zx := by |
| change extend e _ x.val = _ |
| rw [← hzx] |
| exact extend_apply e he.injective _ zx |
| have hfy : f n y = scalar (fun i => e ⁻¹' ball (p (N n) i) (2*r)) (ρ n) (m n) (hm n) zy := by |
| change extend e _ y.val = _ |
| rw [← hzy] |
| exact extend_apply e he.injective _ zy |
| change (f n x-f n y)^2 ≤ _ |
| rw [hfx,hfy] |
| exact hmain |
|
|
| end CannonApprox |
|
|
| open Set Filter Topology TopologicalSpace Metric |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
