Additive combinatorics

Definition

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

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

Remarks

The Cauchy–Davenport theorem is a direct result: in $\Z/p\Z$ it gives $\abs{A + B} \ge \min(p, \abs{A} + \abs{B} - 1)$ . Freiman's theorem is an inverse result: sets of integers with small doubling lie in small generalized arithmetic progressions. Szemerédi's theorem 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.

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Abelian group concept
Sumset concept
Additive combinatorics definition

Tag	`P285NBAVA3`
URI	`m2://P285NBAVA3`
Kind	definition
Writing	W0 stub
MSC	11B30, 11P70, 05D99
Defines	Additive combinatorics
Uses	Sumset, Abelian group
Changed	2 November 2026

Additive combinatorics

Definition

Remarks

Reading path

Further reading

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