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465b10d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped BigOperators Topology
universe u
namespace SubpolynomialLp
def coordinateDistance (p : ℝ) {d : ℕ} (x y : Fin d → ℝ) : ℝ :=
(∑ a, |x a - y a| ^ p) ^ (1 / p)
/-- The universal property used in the definition of d_p(n,D). -/
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‖
/-- The least universal coordinate dimension. The theorem also asserts that
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}
def gamma (p : ℝ) : ℝ := if p < 2 then 2 - p else 1 - 2 / p
/-- The universal coordinate dimension bounds and their logarithmic limits. -/
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
end SubpolynomialLp
end
end OAI
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