Additive number theory

Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Two principal objects of study are the sumset of two subsets A and B of elements from an abelian group G,

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

and the h-fold sumset of A,

${\displaystyle hA={\underset {h}{\underbrace {A+\cdots +A} }}\,.}$