---
tag: 3NM362A2JG
kind: concept
name: Additive combinatorics
msc: [11B30, 11P70, 05D99]
sources:
  - url: https://en.wikipedia.org/wiki/Additive_combinatorics
same_as:
  wikidata: Q56313053
  wikipedia: Additive combinatorics
---

## Summary

Additive combinatorics studies finite subsets of abelian groups, such as sets of integers, and how they behave under addition.
Typical questions ask how large a [[sumset|XNBN31HQWB]] $A + B$ must be, and what a set with a small sumset looks like.
Others ask when a set contains additive patterns such as arithmetic progressions.
It combines combinatorics, Fourier analysis, ergodic theory and number theory.
Its results include [[Szemerédi's theorem|B23B0QEMN5]], [[Freiman's theorem|KEPFYE6WYZ]] and the [[Green–Tao theorem|H8BAHWSEMC]].

## Explanations

### L1 human

Take a finite set of numbers and add every pair of them, allowing a number to be added to itself.
Additive combinatorics asks how many different sums you get, and what that number tells you about the set.
For example, $\set{1, 2, 3, 4}$ gives only the sums $2, \dots, 8$, seven values, because it is very structured.
A random-looking set like $\set{1, 2, 5, 11}$ gives ten different sums.
The field also asks when a large set must contain evenly spaced numbers.