Documentation

Mathlib.MeasureTheory.Group.GeometryOfNumbers

Geometry of numbers #

In this file we prove some of the fundamental theorems in the geometry of numbers, as studied by Hermann Minkowski.

Main results #

TODO #

References #

theorem MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd {E : Type u_1} {L : Type u_2} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {F : Set E} {s : Set E} [AddCommGroup L] [Countable L] [AddAction L E] [MeasurableSpace L] [MeasurableVAdd L E] [MeasureTheory.VAddInvariantMeasure L E μ] (fund : MeasureTheory.IsAddFundamentalDomain L F μ) (hS : MeasureTheory.NullMeasurableSet s μ) (h : μ F < μ s) :
∃ (x : L) (y : L), x y ¬Disjoint (x +ᵥ s) (y +ᵥ s)

Blichfeldt's Theorem. If the volume of the set s is larger than the covolume of the countable subgroup L of E, then there exist two distinct points x, y ∈ L such that (x + s) and (y + s) are not disjoint.

theorem MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {F : Set E} {s : Set E} [NormedAddCommGroup E] [NormedSpace E] [BorelSpace E] [FiniteDimensional E] [MeasureTheory.Measure.IsAddHaarMeasure μ] {L : AddSubgroup E} [Countable L] (fund : MeasureTheory.IsAddFundamentalDomain (L) F μ) (h_symm : xs, -x s) (h_conv : Convex s) (h : μ F * 2 ^ FiniteDimensional.finrank E < μ s) :
∃ (x : L), x 0 x s

The Minkowski Convex Body Theorem. If s is a convex symmetric domain of E whose volume is large enough compared to the covolume of a lattice L of E, then it contains a non-zero lattice point of L.

theorem MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure {E : Type u_1} [MeasurableSpace E] {μ : MeasureTheory.Measure E} {F : Set E} {s : Set E} [NormedAddCommGroup E] [NormedSpace E] [BorelSpace E] [FiniteDimensional E] [Nontrivial E] [MeasureTheory.Measure.IsAddHaarMeasure μ] {L : AddSubgroup E} [Countable L] [DiscreteTopology L] (fund : MeasureTheory.IsAddFundamentalDomain (L) F μ) (h_symm : xs, -x s) (h_conv : Convex s) (h_cpt : IsCompact s) (h : μ F * 2 ^ FiniteDimensional.finrank E μ s) :
∃ (x : L), x 0 x s

The Minkowski Convex Body Theorem for compact domain. If s is a convex compact symmetric domain of E whose volume is large enough compared to the covolume of a lattice L of E, then it contains a non-zero lattice point of L. Compared to exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure, this version requires in addition that s is compact and L is discrete but provides a weaker inequality rather than a strict inequality.