# mathlib3documentation

measure_theory.group.geometry_of_numbers

# Geometry of numbers #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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
• 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 #

theorem measure_theory.exists_pair_mem_lattice_not_disjoint_vadd {E : Type u_1} {L : Type u_2} {μ : measure_theory.measure E} {F s : set E} [countable L] [ E] [ E] (fund : μ) (hS : μ) (h : μ F < μ s) :
(x 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 exists two distincts points x, y ∈ L such that (x + s) and (y + s) are not disjoint.

theorem measure_theory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure {E : Type u_1} {μ : measure_theory.measure E} {F s : set E} [ E] [borel_space E] {L : add_subgroup E} [countable L] (fund : μ) (h : μ F * 2 ^ fdim E < μ s) (h_symm : (x : E), x s -x s) (h_conv : s) :
(x : L) (H : x 0), x s

The Minkowksi 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.