# 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 #

• exists_pair_mem_lattice_not_disjoint_vadd: Blichfeldt's principle, existence of two distinct points in a subgroup such that the translates of a set by these two points are not disjoint when the covolume of the subgroup is larger than the volume of the set.
• exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure: Minkowski's theorem, existence of a non-zero lattice point inside a convex symmetric domain of large enough volume.

## References #

• [Pete L. Clark, Geometry of Numbers with Applications to Number Theory][clark_gon] p.28
theorem MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd {E : Type u_1} {L : Type u_2} [] {μ : } {F : Set E} {s : Set E} [] [] [] [] [] (fund : ) (hS : ) (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} [] {μ : } {F : Set E} {s : Set E} [] [] [] {L : } [] (fund : ) (h_symm : xs, -x s) (h_conv : ) (h : μ F * < 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} [] {μ : } {F : Set E} {s : Set E} [] [] [] [] {L : } [] [] (fund : ) (h_symm : xs, -x s) (h_conv : ) (h_cpt : ) (h : μ F * 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.