| import OAI.Geometry.LatticeCovering.Assembly | |
| namespace OAI | |
| /-! A uniform, unconditional single-lattice covering bound in every dimension at least two. | |
| The eventual bound uses the proved analytic and geometric producers; finite-dimensional | |
| absorption supplies one absolute constant for all dimensions. -/ | |
| noncomputable section | |
| namespace SingleLatticeCovering | |
| open MeasureTheory | |
| open scoped Pointwise | |
| 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 | |
| obtain ⟨C,hC,h⟩ := Assembly.eventual_single_lattice_covering | |
| obtain ⟨C',hC',h'⟩ := FiniteDimensions.absorb_finitely_many_dimensions ⟨C,h⟩ | |
| exact ⟨C',hC',fun n hn K hK => h' n hn K hK.1 hK.2.1 hK.2.2⟩ | |
| end SingleLatticeCovering | |
| end | |
| end OAI | |