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.
- Calculate the volume of the fundamental domain of a finite index subgroup
- Voronoi diagrams
- See Pete L. Clark, Abstract Geometry of Numbers: Linear Forms (arXiv) for some more ideas.
- [Pete L. Clark, Geometry of Numbers with Applications to Number Theory][clark_gon] p.28
Blichfeldt's Theorem. If the volume of the set
s is larger than the covolume of the
E, then there exist two distinct points
x, y ∈ L such that
(x + s)
(y + s) are not disjoint.
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
E, then it contains a non-zero
lattice point of