| import OAI.NumberTheory.DukePrimeDegree.Dynamics.ShearingTimeSelection |
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| namespace OAI |
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| /-! |
| # Diagonal orbit control for both shearing target sets |
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| The same large measurable set controls visits to the compact-return good |
| set and the small-leaf-ball good set. Together with the common-time lemma |
| this supplies the two-target recurrence step of EKL Lemmas 4.5 and 4.6. |
| -/ |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ENNReal |
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| namespace DukePrimeDegree |
| open PrimeDegreePackets |
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| theorem exists_diagonal_two_target_control |
| (d : ℕ) (j : Fin (d + 1)) (μ : Measure (X (d + 1))) [IsFiniteMeasure μ] |
| (hA : ∀ u : LogCoordinates d, |
| μ.map (fun x => rightAction x (diagonalFlow u)) = μ) |
| (G H : Set (X (d + 1))) (hG : MeasurableSet G) (hH : MeasurableSet H) |
| (ε : ℝ) (hε : 0 < ε) : |
| ∃ Z : Set (X (d + 1)), MeasurableSet Z ∧ |
| μ Zᶜ ≤ ENNReal.ofReal (2 * μ.real Gᶜ / ε) + ENNReal.ofReal (2 * μ.real Hᶜ / ε) ∧ |
| ∀ z ∈ Z, ∀ T : ℝ, 1 ≤ T → |
| volume.real (Ioc 0 T ∩ |
| {t | rightAction z (rootCanonicalFlow d j (-2 * t)) ∉ G}) ≤ ε * T ∧ |
| volume.real (Ioc 0 T ∩ |
| {t | rightAction z (rootCanonicalFlow d j (-2 * t)) ∉ H}) ≤ ε * T := by |
| obtain ⟨ZG, hZG, hmG, htG⟩ := |
| exists_diagonal_return_good_set d j μ hA Gᶜ hG.compl ε hε |
| obtain ⟨ZH, hZH, hmH, htH⟩ := |
| exists_diagonal_return_good_set d j μ hA Hᶜ hH.compl ε hε |
| refine ⟨ZG ∩ ZH, hZG.inter hZH, ?_, ?_⟩ |
| · rw [compl_inter] |
| exact (measure_union_le _ _).trans (add_le_add hmG hmH) |
| · intro z hz T hT |
| exact ⟨htG z hz.1 T hT, htH z hz.2 T hT⟩ |
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| theorem common_diagonal_two_scale_time |
| (d : ℕ) (j : Fin (d + 1)) (G H : Set (X (d + 1))) |
| (z : Fin 2 → X (d + 1)) (T ε : ℝ) (hT : 1 ≤ T) (hε : 36 * ε < 1) |
| (θ η : Fin 2 → ℝ) |
| (hθlo : ∀ q, 4 * T ≤ θ q) (hθhi : ∀ q, θ q ≤ 6 * T) |
| (hη : ∀ q, η q = 0 ∨ η q = 1) |
| (hcontrol : ∀ k, ∀ S : ℝ, 1 ≤ S → |
| volume.real (Ioc 0 S ∩ |
| {t | rightAction (z k) (rootCanonicalFlow d j (-2 * t)) ∉ G}) ≤ ε * S ∧ |
| volume.real (Ioc 0 S ∩ |
| {t | rightAction (z k) (rootCanonicalFlow d j (-2 * t)) ∉ H}) ≤ ε * S) : |
| ∃ τ ∈ Ioc 0 T, |
| (∀ k, rightAction (z k) (rootCanonicalFlow d j (-2 * τ)) ∈ G) ∧ |
| ∀ k q, rightAction (z k) |
| (rootCanonicalFlow d j (-(θ q + (2 - η q) * τ))) ∈ H := by |
| obtain ⟨τ, hτ, hG, hH⟩ := exists_common_two_scale_time |
| (fun t x => rightAction x (rootCanonicalFlow d j (-2 * t))) G H z T ε |
| (by linarith) hε θ η hθlo hθhi hη |
| (fun k => (hcontrol k T hT).1) |
| (fun k => (hcontrol k (4 * T) (by linarith)).2) |
| refine ⟨τ, hτ, hG, ?_⟩ |
| intro k q |
| have he : -2 * ((θ q + (2 - η q) * τ) / 2) = -(θ q + (2 - η q) * τ) := by ring |
| simpa only [he] using hH k q |
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| end DukePrimeDegree |
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| end |
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| end OAI |
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