---
tag: P285NBAVA3
kind: definition
name: Additive combinatorics
msc: [11B30, 11P70, 05D99]
defines: 3NM362A2JG
uses: [XNBN31HQWB, 082GHNPJ53]
sources:
  - url: https://en.wikipedia.org/wiki/Additive_combinatorics
---

## Definition

Additive combinatorics is the part of mathematics that studies finite subsets $A, B$ of an [[abelian group|082GHNPJ53]] $G$ through their additive structure.
Its main objects are the [[sumsets|XNBN31HQWB]] $A + B$, the difference sets $A - B$, and the solutions of linear equations with all variables in $A$.
Its typical results are of three kinds:

1. Direct results bound the size of $A + B$ from below in terms of $\abs{A}$ and $\abs{B}$.
2. Inverse results describe the structure of $A$ when $\abs{A + A}$ is small compared with $\abs{A}$.
3. Density results show that large subsets of $G$ contain solutions of given linear equations.

## Remarks

The [[Cauchy–Davenport theorem|F7K7XFACSG]] is a direct result: in $\Z/p\Z$ it gives $\abs{A + B} \ge \min(p, \abs{A} + \abs{B} - 1)$.
[[Freiman's theorem|KEPFYE6WYZ]] is an inverse result: sets of integers with small doubling lie in small generalized arithmetic progressions.
[[Szemerédi's theorem|B23B0QEMN5]] is a density result about arithmetic progressions.
A statement about sets without any addition, such as a bound on chains in a poset, is not a result of additive combinatorics.
The field is also called arithmetic combinatorics.