# Arithmetic combinatorics

In mathematics, arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive combinatorics is the special case when only the operations of addition and subtraction are involved.

For example: if A is a set of N integers, how large or small can the sumset

$A+A := \{x+y: x,y \in A\},$

the difference set

$A-A := \{x-y: x,y \in A\},$

and the product set

$A\times A := \{xy: x,y \in A\}$

be, and how are the sizes of these sets related? (Not to be confused: the terms difference set and product set can have other meanings.)

The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.[1]

Arithmetic combinatorics is explained in Green's review of "Additive Combinatorics" by Tao and Vu.