---
tag: 48EA9Z10DX
kind: definition
name: Sumset
msc: [11B13, 11P70]
defines: XNBN31HQWB
uses: [082GHNPJ53]
sources:
  - url: https://en.wikipedia.org/wiki/Minkowski_addition
---

## Definition

Let $G$ be an [[abelian group|082GHNPJ53]] written additively, and let $A$ and $B$ be subsets of $G$.
The *sumset* of $A$ and $B$ is
$$A + B = \set{a + b : a \in A, \ b \in B}.$$
The *difference set* is $A - B = \set{a - b : a \in A, \ b \in B}$.
For an integer $k \ge 1$, the iterated sumset $kA$ is $A + \dots + A$ with $k$ summands.

## Remarks

In $\Z$, take $A = \set{0, 1, 2}$ and $B = \set{0, 5}$.
Then $A + B = \set{0, 1, 2, 5, 6, 7}$, since each sum $a + b$ lies in this set and each element of it is such a sum.
The set $\set{a + a : a \in A} = \set{0, 2, 4}$ is not the sumset $A + A$, because it misses sums of two different elements such as $0 + 1$.
The iterated sumset $2A = A + A$ is different from the dilate $2 \cdot A = \set{2a : a \in A}$.
For finite nonempty $A, B \subseteq \Z$ one has $\abs{A + B} \ge \abs{A} + \abs{B} - 1$, with equality for arithmetic progressions of the same common difference.
For subsets of $\R^n$ the sumset is called the Minkowski sum.