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Sumset - Wikipedia

  • ️Tue Dec 06 2022

From Wikipedia, the free encyclopedia

In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets {\displaystyle A} and {\displaystyle B} of an abelian group {\displaystyle G} (written additively) is defined to be the set of all sums of an element from {\displaystyle A} with an element from {\displaystyle B}. That is,

{\displaystyle A+B=\{a+b:a\in A,b\in B\}.}

The {\displaystyle n}-fold iterated sumset of {\displaystyle A} is

{\displaystyle nA=A+\cdots +A,}

where there are {\displaystyle n} summands.

Many of the questions and results of additive combinatorics and additive number theory can be phrased in terms of sumsets. For example, Lagrange's four-square theorem can be written succinctly in the form

{\displaystyle 4\,\Box =\mathbb {N} ,}

where {\displaystyle \Box } is the set of square numbers. A subject that has received a fair amount of study is that of sets with small doubling, where the size of the set {\displaystyle A+A} is small (compared to the size of {\displaystyle A}); see for example Freiman's theorem.