| import Mathlib | |
| namespace OAI | |
| /-! Convex bodies and exact coverings by a single lattice in real coordinate space. -/ | |
| namespace SingleLatticeCovering | |
| open MeasureTheory | |
| open scoped Pointwise | |
| abbrev Space (n : ℕ) := Fin n → ℝ | |
| def IsConvexBody {n : ℕ} (K : Set (Space n)) : Prop := | |
| IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty | |
| def LatticeCovers {n : ℕ} (K : Set (Space n)) (L : Submodule ℤ (Space n)) : Prop := | |
| K + (L : Set (Space n)) = Set.univ | |
| theorem single_lattice_covering : | |
| ∃ C : ℝ, 0 < C ∧ ∀ (n : ℕ), 2 ≤ n → | |
| ∀ (K : Set (Space n)), IsConvexBody K → | |
| ∃ (L : Submodule ℤ (Space n)) (_ : DiscreteTopology L), | |
| IsZLattice ℝ L ∧ LatticeCovers K L ∧ | |
| (volume K).toReal / ZLattice.covolume L ≤ C * (n : ℝ) * Real.log (n : ℝ) := by sorry | |
| end SingleLatticeCovering | |
| end OAI | |