| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter |
| open scoped BigOperators Topology |
| universe u |
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| namespace SubpolynomialLp |
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| def coordinateDistance (p : ℝ) {d : ℕ} (x y : Fin d → ℝ) : ℝ := |
| (∑ a, |x a - y a| ^ p) ^ (1 / p) |
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| / |
| def GoodDimension (p : ℝ) (n : ℕ) (D : ℝ) (d : ℕ) : Prop := |
| ∀ (Ω : Type u) (mΩ : MeasurableSpace Ω) (μ : @Measure Ω mΩ) |
| (x : Fin n → @Lp Ω ℝ mΩ _ (ENNReal.ofReal p) μ), |
| Function.Injective x → |
| ∃ (y : Fin n → Fin d → ℝ) (s : ℝ), 0 < s ∧ |
| ∀ i j, s * ‖x i - x j‖ ≤ coordinateDistance p (y i) (y j) ∧ |
| coordinateDistance p (y i) (y j) ≤ D * s * ‖x i - x j‖ |
|
|
| / |
| this least dimension has the defining property, ruling out an empty infimum. -/ |
| def dimension (p : ℝ) (n : ℕ) (D : ℝ) : ℕ := |
| sInf {d : ℕ | GoodDimension.{u} p n D d} |
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| def gamma (p : ℝ) : ℝ := if p < 2 then 2 - p else 1 - 2 / p |
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| / |
| theorem source_main (p : ℝ) (hp : 1 < p) (hp2 : p ≠ 2) : |
| (∀ D : ℝ, 1 < D → ∃ C : ℝ, ∀ n : ℕ, 2 ≤ n → |
| GoodDimension.{u} p n D (dimension.{u} p n D) ∧ |
| Real.log (n : ℝ) / Real.log (1 + 2 * D) ≤ (dimension.{u} p n D : ℝ) ∧ |
| (dimension.{u} p n D : ℝ) ≤ Real.exp (C * Real.log (n : ℝ) ^ gamma p)) ∧ |
| (∀ n : ℕ, 9 ≤ n → |
| GoodDimension.{u} p n 1 (dimension.{u} p n 1) ∧ |
| ((n - 1) / 4) ^ 2 ≤ dimension.{u} p n 1 ∧ |
| dimension.{u} p n 1 ≤ n.choose 2) ∧ |
| (∀ D : ℝ, 1 ≤ D → |
| Tendsto (fun n : ℕ => Real.log (dimension.{u} p n D : ℝ) / Real.log (n : ℝ)) |
| atTop (𝓝 (if 1 < D then 0 else 2))) := by |
| sorry |
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| end SubpolynomialLp |
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| end |
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| end OAI |
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|
