| import OAI.MeasureTheory.DyadicAvoidance.PeriodicBasic |
| import OAI.MeasureTheory.DyadicAvoidance.AdaptiveLabels |
| import OAI.MeasureTheory.DyadicAvoidance.AddressInjectivity |
| import OAI.MeasureTheory.DyadicAvoidance.CenterExposure |
| import OAI.MeasureTheory.DyadicAvoidance.CenterFailureBound |
| import OAI.MeasureTheory.DyadicAvoidance.CenterFirstDefault |
| import OAI.MeasureTheory.DyadicAvoidance.CenterNoDefaultMass |
| import OAI.MeasureTheory.DyadicAvoidance.CenterRouteExposure |
| import OAI.MeasureTheory.DyadicAvoidance.ChildDistribution |
| import OAI.MeasureTheory.DyadicAvoidance.ConcreteLabelMeasures |
| import OAI.MeasureTheory.DyadicAvoidance.ConcreteWindowAtom |
| import OAI.MeasureTheory.DyadicAvoidance.Construction |
| import OAI.MeasureTheory.DyadicAvoidance.DyadicRoutedAddresses |
| import OAI.MeasureTheory.DyadicAvoidance.ExceptionalCenters |
| import OAI.MeasureTheory.DyadicAvoidance.ExceptionalExpectation |
| import OAI.MeasureTheory.DyadicAvoidance.ExpectedRepair |
| import OAI.MeasureTheory.DyadicAvoidance.ExposureMoments |
| import OAI.MeasureTheory.DyadicAvoidance.ExposureUnionBound |
| import OAI.MeasureTheory.DyadicAvoidance.FiniteLocalHit |
| import OAI.MeasureTheory.DyadicAvoidance.FiniteLocalPredicate |
| import OAI.MeasureTheory.DyadicAvoidance.FiniteTableFirstDefault |
| import OAI.MeasureTheory.DyadicAvoidance.FiniteTableModel |
| import OAI.MeasureTheory.DyadicAvoidance.FirstDefault |
| import OAI.MeasureTheory.DyadicAvoidance.FirstDefaultSelection |
| import OAI.MeasureTheory.DyadicAvoidance.FixedScaleProbability |
| import OAI.MeasureTheory.DyadicAvoidance.GlobalFromPeriodic |
| import OAI.MeasureTheory.DyadicAvoidance.GridGeometry |
| import OAI.MeasureTheory.DyadicAvoidance.GridMeasure |
| import OAI.MeasureTheory.DyadicAvoidance.GridModelStability |
| import OAI.MeasureTheory.DyadicAvoidance.GridSeparation |
| import OAI.MeasureTheory.DyadicAvoidance.GridStability |
| import OAI.MeasureTheory.DyadicAvoidance.GridStabilityFin |
| import OAI.MeasureTheory.DyadicAvoidance.GridWindowStability |
| import OAI.MeasureTheory.DyadicAvoidance.LocalPredicateFactor |
| import OAI.MeasureTheory.DyadicAvoidance.MonotoneRepresentatives |
| import OAI.MeasureTheory.DyadicAvoidance.OrderedRoutePreservation |
| import OAI.MeasureTheory.DyadicAvoidance.OrderedRouting |
| import OAI.MeasureTheory.DyadicAvoidance.PackedWindowSpan |
| import OAI.MeasureTheory.DyadicAvoidance.PeriodicEnlargement |
| import OAI.MeasureTheory.DyadicAvoidance.PeriodicEnvelope |
| import OAI.MeasureTheory.DyadicAvoidance.PeriodicMeasure |
| import OAI.MeasureTheory.DyadicAvoidance.PreorderIndex |
| import OAI.MeasureTheory.DyadicAvoidance.RandomSetDensity |
| import OAI.MeasureTheory.DyadicAvoidance.RawRepair |
| import OAI.MeasureTheory.DyadicAvoidance.RepresentativeFailure |
| import OAI.MeasureTheory.DyadicAvoidance.RoutedAddressInjectivity |
| import OAI.MeasureTheory.DyadicAvoidance.RoutedExposure |
| import OAI.MeasureTheory.DyadicAvoidance.RoutedFirstDefault |
| import OAI.MeasureTheory.DyadicAvoidance.RoutedSetDensity |
| import OAI.MeasureTheory.DyadicAvoidance.RoutingBudget |
| import OAI.MeasureTheory.DyadicAvoidance.RoutingPath |
| import OAI.MeasureTheory.DyadicAvoidance.SelectedCoordinateLaw |
| import OAI.MeasureTheory.DyadicAvoidance.SelectorMoment |
| import OAI.MeasureTheory.DyadicAvoidance.StableKeyRouting |
| import OAI.MeasureTheory.DyadicAvoidance.StableRoutingBridge |
| import OAI.MeasureTheory.DyadicAvoidance.TreePreorder |
| import OAI.MeasureTheory.DyadicAvoidance.TreePreorderRank |
| import OAI.MeasureTheory.DyadicAvoidance.TreeSpanBridge |
| import OAI.MeasureTheory.DyadicAvoidance.TrialAddresses |
| import OAI.MeasureTheory.DyadicAvoidance.WindowBudgets |
| import OAI.MeasureTheory.DyadicAvoidance.WindowData |
| import OAI.MeasureTheory.DyadicAvoidance.WindowDataExistence |
| import OAI.MeasureTheory.DyadicAvoidance.WindowDataStability |
| import OAI.MeasureTheory.DyadicAvoidance.WindowExceptionalTransfer |
| import OAI.MeasureTheory.DyadicAvoidance.WindowFixedScale |
| import OAI.MeasureTheory.DyadicAvoidance.WindowLocalFactor |
| import OAI.MeasureTheory.DyadicAvoidance.WindowLocalHit |
| import OAI.MeasureTheory.DyadicAvoidance.WindowMetadata |
| import OAI.MeasureTheory.DyadicAvoidance.WindowPlacement |
| import OAI.MeasureTheory.DyadicAvoidance.WindowRecursion |
| import OAI.MeasureTheory.DyadicAvoidance.WindowSpanSum |
| import OAI.MeasureTheory.DyadicAvoidance.WindowTableModel |
| import OAI.MeasureTheory.DyadicAvoidance.WindowTrialAddresses |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem310 |
|
|
| theorem periodic_hitting_set : |
| ∀ p : ℝ, 0 < p → p < 1 → ∃ H : Set ℝ, IsOpen H ∧ OnePeriodic H ∧ periodDensity H ≤ ENNReal.ofReal (6 * p) ∧ ∀ x t : ℝ, t ∈ Set.Icc (1 : ℝ) 2 → ∃ n : ℕ, 1 ≤ n ∧ x + t * dyadicPoint n ∈ H := by |
| intro p hp hp1 |
| obtain ⟨M, d, R, hM, hd, hR, _, hsurvival, hbudget⟩ := |
| RoutingBudget.exists_nondefault_parameters p hp hp1 |
| obtain ⟨g, hg, hgeo⟩ := exists_windowData_selector_stability M d p hp |
| let W := WindowConstruction.windowData (M + 1) d g R (by omega) hR |
| obtain ⟨G, hG, hGp, hbad, hstable⟩ := hgeo R W |
| apply Construction.periodic_hitting_of_window_failure W p hp hp1 G hG hbad |
| intro x hx |
| apply CenterFailureBound.raw_failure_le_of_outcome_center_atoms |
| (FiniteTableModel.selectorEndpoints W) (FiniteTableModel.terminalEndpoints W) |
| p hp.le hp1.le |
| (RoutedSetDensity.routedSet (FiniteTableModel.selectorEndpoints W) |
| (FiniteTableModel.terminalEndpoints W)) (Set.Ici 1) x |
| (RoutingBudget.ennreal_survival_lt hp M d hsurvival).le |
| intro ξ hξ |
| apply ConcreteWindowAtom.raw_failure_on_good_atom W hM p hp.le hp1.le |
| (fun r hr N hN => (hbudget r hr N hN).le) x ?_ ξ hξ |
| intro P i n hn t ht ω |
| exact (hstable x hx P i n hn t ht ω).symm |
|
|
| theorem dyadic_affine_avoidance : |
| ∀ η : ℝ, 0 < η → η < 1 → ∃ E : Set ℝ, E ⊆ Set.Icc (0 : ℝ) 1 ∧ IsCompact E ∧ MeasureTheory.volume E > ENNReal.ofReal (1 - η) ∧ ∀ x s : ℝ, s ≠ 0 → ∃ n : ℕ, 1 ≤ n ∧ x + s * dyadicPoint n ∉ E := by |
| exact Problem310Support.avoidance_of_periodic_hitting periodic_hitting_set |
|
|
| end Problem310 |
|
|
| end |
|
|
| end OAI |
|
|
