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59d1c80 | 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 | 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
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