| import Mathlib |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace CurrentProjection |
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| abbrev Vector (d : ℕ) := EuclideanSpace ℝ (Fin d) |
| abbrev Sphere (d : ℕ) := Metric.sphere (0 : Vector d) 1 |
| abbrev Rows (k d : ℕ) := Fin k → Vector d |
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| def uniformSphere (d : ℕ) : Measure (Sphere d) := |
| let σ := (volume : Measure (Vector d)).toSphere |
| (σ Set.univ)⁻¹ • σ |
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| def gaussianRows (k d : ℕ) : Measure (Rows k d) := |
| Measure.pi fun _ => stdGaussian (Vector d) |
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| def label {k d : ℕ} (g : Rows k d) (s : Vector d) : Fin k → ℝ := |
| fun i => ⟪s, g i⟫ |
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| def LocalGrowth {d : ℕ} (ρ : Measure (Sphere d)) (a : ℕ) (B : ℝ) : Prop := |
| ∀ z : Vector d, ∀ t : ℝ, 0 < t → |
| ρ {u | dist (u : Vector d) z ≤ t} ≤ ENNReal.ofReal (B * t ^ a) |
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| def cubeAverage {k d : ℕ} (ρ : Measure (Sphere d)) (g : Rows k d) |
| (y : Fin k → ℝ) (δ : ℝ) : ℝ≥0∞ := |
| ENNReal.ofReal ((2 * δ)⁻¹ ^ k) * |
| ρ {u | ∀ i, |label g (u : Vector d) i - y i| ≤ δ} |
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| def exactDensity {k d : ℕ} (ρ : Measure (Sphere d)) (g : Rows k d) |
| (y : Fin k → ℝ) : ℝ≥0∞ := by |
| classical |
| let f := fun j : ℕ => (cubeAverage ρ g y (1 / (j : ℝ))).toReal |
| exact if h : ∃ L : ℝ, Tendsto f atTop (𝓝 L) then ENNReal.ofReal h.choose else 0 |
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| def AdmissibleDimensions (d m r q : ℕ) : Prop := |
| 1 ≤ m ∧ 1 ≤ r ∧ 1 ≤ q ∧ m + r < d - 1 ∧ m + r - m - (q - 1) ≥ 1 |
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| def momentConstant (C : ℝ) (d : ℕ) (B D : ℝ) : ℝ≥0∞ := |
| ENNReal.ofReal (Real.exp (C * d) * B * (1 + max (Real.log (D / B)) 0)) |
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| def MixedMomentMain : Prop := |
| ∃ C : ℝ, 0 < C ∧ ∀ (d m r q : ℕ), AdmissibleDimensions d m r q → |
| ∀ (ρ : Measure (Sphere d)), IsFiniteMeasure ρ → |
| ∀ (B D : ℝ), 0 < B → 0 < D → LocalGrowth ρ (m+r) B → LocalGrowth ρ (d-1) D → |
| (∀ (s : Sphere d) (δ : ℝ), 0 < δ → |
| (∫⁻ X, (∫⁻ Z, cubeAverage ρ (Fin.append X Z) |
| (label (Fin.append X Z) (s : Vector d)) δ ∂gaussianRows r d)^q |
| ∂gaussianRows m d) ≤ (momentConstant C d B D)^q) ∧ |
| (ρ ≪ uniformSphere d → ∃ E : Set (Sphere d), MeasurableSet E ∧ ρ E = 0 ∧ |
| ∀ s ∉ E, |
| (∀ᵐ G ∂gaussianRows (m+r) d, |
| Tendsto (fun δ : ℝ => cubeAverage ρ G (label G (s : Vector d)) δ) |
| (𝓝[>] 0) (𝓝 (exactDensity ρ G (label G (s : Vector d))))) ∧ |
| ((∫⁻ X, (∫⁻ Z, exactDensity ρ (Fin.append X Z) |
| (label (Fin.append X Z) (s : Vector d)) ∂gaussianRows r d)^q |
| ∂gaussianRows m d) ≤ (momentConstant C d B D)^q) ∧ |
| (ρ ≠ 0 → ∀ᵐ G ∂gaussianRows (m+r) d, |
| 0 < exactDensity ρ G (label G (s : Vector d)) ∧ |
| exactDensity ρ G (label G (s : Vector d)) < ∞)) |
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| end CurrentProjection |
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| end |
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| end OAI |
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