---
tag: XNBN31HQWB
kind: concept
name: Sumset
msc: [11B13, 11P70]
sources:
  - url: https://en.wikipedia.org/wiki/Minkowski_addition
---

## Summary

The sumset $A + B$ of two subsets of an abelian group is the set of all sums $a + b$ with $a \in A$ and $b \in B$.
It is the basic object of [[additive combinatorics|3NM362A2JG]].
Comparing $\abs{A + B}$ with $\abs{A}$ and $\abs{B}$ measures how much additive structure the sets have.
For subsets of a vector space the same operation is called Minkowski addition.

## Explanations

### L1 human

Take two sets of numbers and write down every sum of one number from the first set and one from the second.
The collection of these sums is the sumset.
For example, $\set{0, 1} + \set{0, 10} = \set{0, 1, 10, 11}$.
Sets that are evenly spaced have few different sums, while scattered sets have many.