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785c727 | 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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import Mathlib
namespace OAI
namespace CoveringOrder
open Set MeasureTheory
open scoped ENNReal
abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)
/-- Actual discrete full-rank lattice coverings; no assumed density theorem. -/
structure LatticeCover {n : ℕ} (K : Set (Space n)) where
lattice : Submodule ℤ (Space n)
discrete : DiscreteTopology lattice
fullrank : IsZLattice ℝ lattice
covers : ∀ y, ∃ l : lattice, y - l.val ∈ K
noncomputable def euclideanVolume (n : ℕ) :
@Measure (EuclideanSpace ℝ (Fin n))
(@WithLp.measurableSpace 2 (Fin n → ℝ)
(@MeasurableSpace.pi (Fin n) (fun _ => ℝ) (fun _ => Real.measurableSpace))) :=
@Measure.map (Fin n → ℝ) (EuclideanSpace ℝ (Fin n))
(@MeasurableSpace.pi (Fin n) (fun _ => ℝ) (fun _ => Real.measurableSpace))
(@WithLp.measurableSpace 2 (Fin n → ℝ)
(@MeasurableSpace.pi (Fin n) (fun _ => ℝ) (fun _ => Real.measurableSpace)))
(@WithLp.toLp 2 (Fin n → ℝ))
(Measure.pi (fun _ : Fin n => @volume ℝ Real.measureSpace))
noncomputable def euclideanCovolume {n : ℕ}
(lattice : Submodule ℤ (EuclideanSpace ℝ (Fin n))) : ℝ≥0∞ :=
addCovolume lattice (EuclideanSpace ℝ (Fin n)) (euclideanVolume n)
noncomputable def latticeDensity {n : ℕ} {K : Set (Space n)}
(cover : LatticeCover K) : ℝ≥0∞ :=
euclideanVolume n K / euclideanCovolume cover.lattice
noncomputable def thetaL {n : ℕ} (K : Set (EuclideanSpace ℝ (Fin n))) : ℝ≥0∞ :=
⨅ cover : LatticeCover K, latticeDensity cover
end CoveringOrder
namespace TranslativeCovering
open Set Filter MeasureTheory
open scoped ENNReal
abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)
def cube (n : ℕ) (R : ℝ) : Set (Space n) :=
{x | ∀ i, -R ≤ x i ∧ x i ≤ R}
def LocallyFiniteCenters {n : ℕ} (X : Set (Space n)) : Prop :=
∀ B : Set (Space n), Bornology.IsBounded B → (X ∩ B).Finite
def Covers {n : ℕ} (K X : Set (Space n)) : Prop :=
∀ y, ∃ x ∈ X, y - x ∈ K
def ConvexBody {n : ℕ} (K : Set (Space n)) : Prop :=
IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty
def CentrallySymmetric {n : ℕ} (K : Set (Space n)) : Prop :=
∃ z, ∀ x, x ∈ K ↔ (2 : ℝ) • z - x ∈ K
noncomputable def upperCenterIntensity {n : ℕ} (X : Set (Space n)) : ℝ≥0∞ :=
Filter.limsup (fun R : ℝ =>
(↑(X ∩ cube n R).ncard : ℝ≥0∞) / ENNReal.ofReal ((2 * R) ^ n)) atTop
noncomputable def thetaT {n : ℕ} (K : Set (Space n)) : ℝ≥0∞ :=
⨅ X : {X : Set (Space n) // LocallyFiniteCenters X ∧ Covers K X},
volume K * upperCenterIntensity X.val
end TranslativeCovering
namespace CoveringOrder
open Set MeasureTheory
open scoped ENNReal
abbrev Body (n : ℕ) := {K : Set (Space n) // TranslativeCovering.ConvexBody K}
abbrev SymmetricBody (n : ℕ) :=
{K : Body n // TranslativeCovering.CentrallySymmetric K.val}
noncomputable def translativeSup (n : ℕ) : ℝ≥0∞ :=
⨆ K : Body n, TranslativeCovering.thetaT K.val
noncomputable def symmetricTranslativeSup (n : ℕ) : ℝ≥0∞ :=
⨆ K : SymmetricBody n, TranslativeCovering.thetaT K.val.val
noncomputable def latticeSup (n : ℕ) : ℝ≥0∞ :=
⨆ K : Body n, thetaL K.val
noncomputable def symmetricLatticeSup (n : ℕ) : ℝ≥0∞ :=
⨆ K : SymmetricBody n, thetaL K.val.val
/-- The finite real comparison scale, viewed in extended nonnegative reals. -/
def orderBounds (c C : ℝ) (n : ℕ) (a : ℝ≥0∞) : Prop :=
ENNReal.ofReal (c * (n : ℝ) * Real.log (n : ℝ)) ≤ a ∧
a ≤ ENNReal.ofReal (C * (n : ℝ) * Real.log (n : ℝ))
/-- All four extremal quantities have common
absolute positive comparison constants. ENNReal bounds imply finiteness rather
than mapping a potentially infinite density to zero through `toReal`. -/
def OptimalOrder : Prop :=
∃ c C : ℝ, 0 < c ∧ 0 < C ∧ ∃ n₀ : ℕ, ∀ n : ℕ, n₀ ≤ n →
orderBounds c C n (translativeSup n) ∧
orderBounds c C n (symmetricTranslativeSup n) ∧
orderBounds c C n (latticeSup n) ∧
orderBounds c C n (symmetricLatticeSup n)
end CoveringOrder
namespace TranslativeCovering
open Set
def MainStatement : Prop :=
∃ c : ℝ, 0 < c ∧ ∃ n₀ : ℕ, ∀ n : ℕ, n₀ ≤ n →
∃ K : Set (Space n), ConvexBody K ∧ CentrallySymmetric K ∧
ENNReal.ofReal (c * (n : ℝ) * Real.log (n : ℝ)) < thetaT K
end TranslativeCovering
end OAI
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