openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /ControlledCutObservation.lean
| import OAI.NumberTheory.DukePrimeDegree.Geometry.ControlledDistanceCuts | |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.ContractingCutItineraries | |
| namespace OAI | |
| /-! # Finite observations from controlled distance cuts | |
| The observation records the membership bits of the selected distance | |
| balls. Equality of observations gives small relative lifts, while the | |
| dyadic estimates imply almost-sure avoidance of all finite cut boundaries. | |
| -/ | |
| noncomputable section | |
| open Set MeasureTheory Filter | |
| open scoped Topology ENNReal | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| def distanceCutObservation {n N : ℕ} (z : Fin N → X n) (c : Fin N → ℝ) | |
| (x : X n) (i : Fin N) : Bool := decide (quotientLogDistance (z i) x < c i) | |
| lemma distanceCutObservation_measurable {n N : ℕ} | |
| (z : Fin N → X n) (c : Fin N → ℝ) : Measurable (distanceCutObservation z c) := by | |
| apply measurable_pi_iff.mpr | |
| intro i | |
| apply measurable_to_bool | |
| have he : (fun x => distanceCutObservation z c x i) ⁻¹' {true} = | |
| {x | quotientLogDistance (z i) x < c i} := by | |
| ext x | |
| simp [distanceCutObservation] | |
| rw [he] | |
| exact measurableSet_lt (continuous_quotientLogDistance n _).measurable measurable_const | |
| lemma distanceCutObservation_small_lifts {n N : ℕ} | |
| (z : Fin N → X n) (c : Fin N → ℝ) {r : ℝ} | |
| (hsmall : ∀ i, ∀ x, quotientLogDistance (z i) x < c i → | |
| ∀ y, quotientLogDistance (z i) y < c i → | |
| ∃ g ∈ matrixBall n r, rightAction x g=y) | |
| {x y : X n} (hx : ∃ i, quotientLogDistance (z i) x < c i) | |
| (hxy : distanceCutObservation z c y=distanceCutObservation z c x) : | |
| ∃ g ∈ matrixBall n r, rightAction x g=y := by | |
| obtain ⟨i,hi⟩ := hx | |
| apply hsmall i x hi y | |
| have H := congrFun hxy i | |
| simpa only [distanceCutObservation,hi,decide_true,decide_eq_true_eq] using H | |
| theorem ae_eventually_avoid_finite_dyadic_boundaries | |
| {Y : Type*} [MeasurableSpace Y] {N : ℕ} | |
| (μ : Measure Y) {T : Y → Y} (hT : MeasurePreserving T μ μ) | |
| (f : Fin N → Y → ℝ) (hf : ∀ i, Measurable (f i)) (c C : Fin N → ℝ) | |
| (hbound : ∀ i k, μ (realBoundaryStrip (f i) (c i) ((1/4:ℝ)^k)) ≤ | |
| ENNReal.ofReal (C i*(1/2:ℝ)^k)) : | |
| ∀ᵐ x ∂μ, ∀ᶠ k : ℕ in atTop, ∀ i, | |
| (1/4:ℝ)^k < |f i (T^[k] x)-c i| := by | |
| have hall (i : Fin N) : ∀ᵐ x ∂μ, ∀ᶠ k : ℕ in atTop, | |
| (1/4:ℝ)^k < |f i (T^[k] x)-c i| := by | |
| apply ae_eventually_avoid_realBoundaryStrip μ hT (hf i) | |
| have hs := ENNReal.tsum_le_tsum (hbound i) | |
| exact (lt_of_le_of_lt hs | |
| (summable_geometric_two.mul_left (C i)).tsum_ofReal_lt_top).ne | |
| filter_upwards [ae_all_iff.mpr hall] with x hx | |
| exact Filter.eventually_all.mpr hx | |
| end DukePrimeDegree | |
| end | |
| end OAI | |