---
tag: AAATH70CY1
kind: concept
name: Geometry of numbers
msc: [11H06, 52C07]
sources:
  - url: https://en.wikipedia.org/wiki/Geometry_of_numbers
same_as:
  wikidata: Q1466519
  wikipedia: Geometry of numbers
---

## Summary

The geometry of numbers studies [[lattices|WN9V7WSJKF]] 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|0K387S2PKQ]] says that a large symmetric convex set contains a nonzero lattice point.
Its main tools include [[Blichfeldt's theorem|A3FTHW4NES]], [[successive minima|YDMBKD8FP2]] and [[Minkowski's second theorem|5H4XZVF9B9]].
It gives results in algebraic number theory, Diophantine approximation and quadratic forms, and it is closely tied to sphere packing.

## Explanations

### L1 human

Many questions about whole numbers can be turned into questions about points of a grid.
The geometry of numbers answers them by looking at shapes and their areas or volumes.
If a shape is large enough and symmetric about the origin, it must contain a grid point other than the origin.
For instance, this idea shows that every prime of the form $4k + 1$ is a sum of two squares.