File size: 2,500 Bytes
cfa2fd2 | 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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators Topology ENNReal
open Classical Filter
namespace TriangularUniversal
abbrev Plane := EuclideanSpace ℝ (Fin 2)
def LocallyFinite (C : Set Plane) : Prop :=
∀ R : ℝ, (C ∩ Metric.closedBall (0 : Plane) R).Finite
def diskPoints (C : Set Plane) (R : ℝ) : Finset Plane :=
if h : (C ∩ Metric.closedBall (0 : Plane) R).Finite then h.toFinset else ∅
def diskCount (C : Set Plane) (R : ℝ) : ℕ := (diskPoints C R).card
def DensityOne (C : Set Plane) : Prop :=
Tendsto (fun R : ℝ => (diskCount C R : ℝ) / (Real.pi * R ^ 2))
atTop (𝓝 1)
def AdmissiblePotential (g : ℝ → ℝ) : Prop :=
ContDiffOn ℝ (⊤ : ℕ∞) g (Set.Ioi 0) ∧
(∀ t : ℝ, 0 < t → 0 ≤ g t) ∧
∀ (r : ℕ) (t : ℝ), 0 < t → 0 ≤ (-1 : ℝ) ^ r * iteratedDeriv r g t
def diskEnergy (g : ℝ → ℝ) (C : Set Plane) (R : ℝ) : ℝ≥0∞ :=
(diskCount C R : ℝ≥0∞)⁻¹ *
∑ x ∈ diskPoints C R, ∑ y ∈ (diskPoints C R).erase x,
ENNReal.ofReal (g (‖x - y‖ ^ 2))
def energy (g : ℝ → ℝ) (C : Set Plane) : ℝ≥0∞ :=
Filter.liminf (diskEnergy g C) atTop
end TriangularUniversal
namespace AtomicTriangular
abbrev Plane := EuclideanSpace ℝ (Fin 2)
def b : ℝ := Real.sqrt 3 / 2
def triangularPoint (jk : ℤ × ℤ) : Plane :=
(Real.sqrt b)⁻¹ •
(EuclideanSpace.single 0 ((jk.1 : ℝ) + (jk.2 : ℝ) / 2) +
EuclideanSpace.single 1 ((jk.2 : ℝ) * b))
def A : Set Plane := Set.range triangularPoint
abbrev LocallyFinite := TriangularUniversal.LocallyFinite
abbrev diskPoints := TriangularUniversal.diskPoints
abbrev diskCount := TriangularUniversal.diskCount
abbrev DensityOne := TriangularUniversal.DensityOne
abbrev AdmissiblePotential := TriangularUniversal.AdmissiblePotential
abbrev diskEnergy := TriangularUniversal.diskEnergy
abbrev energy := TriangularUniversal.energy
/-- The full nonzero-lattice energy, allowing a divergent series. -/
def latticeEnergy (g : ℝ → ℝ) : ℝ≥0∞ :=
∑' a : {x : Plane // x ∈ A ∧ x ≠ 0}, ENNReal.ofReal (g (‖a.val‖ ^ 2))
/-- Universal energy minimality for smooth completely monotone potentials, with attainment. -/
theorem universal_energy_minimum (g : ℝ → ℝ) (C : Set Plane)
(hg : AdmissiblePotential g) (hC : LocallyFinite C) (hd : DensityOne C) :
latticeEnergy g ≤ energy g C ∧ latticeEnergy g = energy g A := by
sorry
end AtomicTriangular
end
end OAI
|