# Arithmetic combinatorics

Arithmetic combinatorics

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 refers to 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 belong to other spaces than the integers. e.g. groups, rings and fields.[1]

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