| import OAI.MeasureTheory.Falconer.Geometry.SelectTransverseChartTriple |
| import OAI.MeasureTheory.Falconer.Slicing.EndpointSliceObstruction |
|
|
| namespace OAI |
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| /-! # Exclusion of common chart incidence by the three-tube estimate -/ |
|
|
| namespace Falconer |
| open Set Metric MeasureTheory |
| open scoped ENNReal |
| variable {n : ℕ} |
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| theorem endpoint_chart_incidence_obstruction |
| (α : Measure (Space (n+1))) (β : Measure (Space n)) |
| [IsFiniteMeasure α] [IsFiniteMeasure β] |
| (V : Submodule ℝ (Space (n+1))) (e : V ≃ₗᵢ[ℝ] Space 3) |
| (E : Set (Space (n+1) × Space n)) (hE : MeasurableSet E) |
| (G : Set (Space n × Space n)) (hG : MeasurableSet G) |
| (J : Space n → Finset (Space (n+1))) |
| {δ η a κ C s R M b : ℝ} |
| (hδ : 0 < δ) (hδ1 : δ ≤ 1) (ha : 0 ≤ a) (hC : 0 ≤ C) (hη : η < 1) |
| (hM : 0 < M) (hb : 0 < b) |
| (hα : α univ ≤ ENNReal.ofReal M) (hβ : β univ ≤ ENNReal.ofReal M) |
| (hchart : ∀ u ∈ E, ‖u.2‖ ≤ 1) |
| (hbounded : ∀ u ∈ E, ‖u.1‖ ≤ R) |
| (hband : ∀ u ∈ E, gnomonicChart u.2 ∈ directionSliceBand V η) |
| (hball : ∀ p : Space 3, α {x | dist (e (V.orthogonalProjectionOnto x)) p < 6*δ} ≤ |
| ENNReal.ofReal (δ^(-κ)*(6*δ)^s)) |
| (hcover : ∀ u ∈ E, ∃ v ∈ J u.2, |
| ‖perpendicularPart (gnomonicChart u.2 : Space (n+1)) (u.1-v)‖ ≤ C*δ) |
| (hcard : ∀ p, ((J p).card:ℝ) ≤ δ^(-1-κ)) |
| (hbad : (β.prod (β.prod β)) (badChartTriples G (2*δ^a))+ |
| 2*β univ*(β.prod β) Gᶜ ≤ ENNReal.ofReal (b^3/(4*M^3))) |
| (hmass : δ^κ ≤ b^3/(2*M^5)) |
| (herror : 72*η ≤ 3*(δ^a)^2) |
| (hbudget : (δ^(-1-κ))^3*((6*C+35)/δ^a)^6*(δ^(-κ)*(6*δ)^s)^2 < (δ^κ)^2) : |
| (α.prod β) E < ENNReal.ofReal b := by |
| classical |
| by_contra! hI |
| obtain ⟨w,hvol,_,_,hcommon⟩ := exists_transverse_common_chart_triple α β E hE hchart |
| G hG (2*δ^a) hM hb hα hβ hI hbad |
| let Q : Set (Space (n+1)) := {x | (x,w) ∈ commonTripleRelation E} |
| let v : Fin 3 → Sphere (n+1) := ![gnomonicChart w.1,gnomonicChart w.2.1,gnomonicChart w.2.2] |
| let L : Fin 3 → Finset (Space (n+1)) := ![J w.1,J w.2.1,J w.2.2] |
| have hQ (x : Space (n+1)) (hx : x ∈ Q) : |
| (x,w.1) ∈ E ∧ (x,w.2.1) ∈ E ∧ (x,w.2.2) ∈ E := hx |
| have hn : Q.Nonempty := by |
| by_contra hn |
| have he : Q = ∅ := not_nonempty_iff_eq_empty.mp hn |
| change ENNReal.ofReal (b^3/(2*M^5)) < α Q at hcommon |
| rw [he,measure_empty] at hcommon |
| exact not_lt_of_ge (bot_le : (0 : ℝ≥0∞) ≤ _) hcommon |
| obtain ⟨x,hx⟩ := hn |
| have hvband : ∀ i, v i ∈ directionSliceBand V η := by |
| intro i |
| fin_cases i |
| · exact hband (x,w.1) (hQ x hx).1 |
| · exact hband (x,w.2.1) (hQ x hx).2.1 |
| · exact hband (x,w.2.2) (hQ x hx).2.2 |
| have hvgram : 2*δ^a ≤ Real.sqrt ((Matrix.gram ℝ (fun i => (v i : Space (n+1)))).det) := hvol |
| have hQbound : ∀ y ∈ Q, ‖y‖ ≤ R := fun y hy => hbounded (y,w.1) (hQ y hy).1 |
| have hQcover : ∀ i : Fin 3, ∀ y ∈ Q, ∃ c ∈ L i, |
| ‖perpendicularPart (v i : Space (n+1)) (y-c)‖ ≤ C*δ := by |
| intro i y hy |
| fin_cases i |
| · exact hcover (y,w.1) (hQ y hy).1 |
| · exact hcover (y,w.2.1) (hQ y hy).2.1 |
| · exact hcover (y,w.2.2) (hQ y hy).2.2 |
| have hLcard : ∀ i, ((L i).card:ℝ) ≤ δ^(-1-κ) := by |
| intro i |
| fin_cases i |
| · exact hcard w.1 |
| · exact hcard w.2.1 |
| · exact hcard w.2.2 |
| have hQmass : δ^κ ≤ (α Q).toReal := hmass.trans (by |
| have hh := ENNReal.toReal_mono (measure_ne_top α Q) hcommon.le |
| simpa only [ENNReal.toReal_ofReal (by positivity : 0 ≤ b^3/(2*M^5))] using hh) |
| exact endpoint_slice_obstruction α V e v hδ hδ1 ha hC hη hvband hvgram herror hbudget |
| Q L R hQbound hball hQcover hLcard hQmass |
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| end Falconer |
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| end OAI |
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