| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter |
| open scoped Topology |
|
|
| namespace WeakPinned |
| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
|
|
| def distanceFiber (P : Finset Plane) (x y : Plane) : Finset Plane := |
| (P.erase x).filter (fun z => dist z x = dist y x) |
|
|
| def k (P : Finset Plane) (x y : Plane) : ℕ := |
| (distanceFiber P x y).card |
|
|
| def richPairs (P : Finset Plane) (s : ℝ) : Finset (Plane × Plane) := |
| (P ×ˢ P).filter (fun xy => xy.1 ≠ xy.2 ∧ |
| (P.card : ℝ) ^ s ≤ (k P xy.1 xy.2 : ℝ)) |
|
|
| def pairFraction (P : Finset Plane) (s : ℝ) : ℝ := |
| (richPairs P s).card / ((P.card : ℝ) * ((P.card : ℝ) - 1)) |
|
|
| def F (n : ℕ) (s : ℝ) : ℝ := |
| if 2 ≤ n then sSup {a : ℝ | ∃ P : Finset Plane, P.card = n ∧ a = pairFraction P s} |
| else 0 |
|
|
| def distances (P : Finset Plane) (x : Plane) : Finset ℝ := |
| (P.erase x).image (fun y => dist y x) |
|
|
| def badPins (P : Finset Plane) (ε : ℝ) : Finset Plane := |
| P.filter (fun x => ((distances P x).card : ℝ) < (P.card : ℝ) ^ (1 - ε)) |
|
|
| def badPinFraction (P : Finset Plane) (ε : ℝ) : ℝ := |
| (badPins P ε).card / (P.card : ℝ) |
|
|
| def B (n : ℕ) (ε : ℝ) : ℝ := |
| if 0 < n then sSup {a : ℝ | ∃ P : Finset Plane, P.card = n ∧ a = badPinFraction P ε} |
| else 0 |
|
|
| /- Uniform multiplicity and pinned-distance convergence statements. -/ |
| def MainStatement : Prop := |
| ∀ s : ℝ, 0 < s → Tendsto (fun n : ℕ => F n s) atTop (nhds 0) |
|
|
| def PinsStatement : Prop := |
| ∀ ε : ℝ, 0 < ε → Tendsto (fun n : ℕ => B n ε) atTop (nhds 0) |
|
|
| end WeakPinned |
|
|
| end |
|
|
| end OAI |
|
|
