Geometry of numbers

Summary

The geometry of numbers studies lattices in $\R^n$ together with convex bodies and other regions, and uses volume arguments to find lattice points in them. It was founded by Hermann Minkowski, whose convex body theorem says that a large symmetric convex set contains a nonzero lattice point. Its main tools include Blichfeldt's theorem, successive minima and Minkowski's second theorem. It gives results in algebraic number theory, Diophantine approximation and quadratic forms, and it is closely tied to sphere packing.