1.NounThe subfield of number theory concerning the behaviour of sumsets (integer sets added to each other elementwise)."Abstractly, additive number theory includes the study of abelian groups and commutative semigroups with an addition operation.; Two principal objects of study in additive number theory are the sumset
A
+
B
=
{
a
+
b
:
a
∈
A
,
b
∈
B
}
{\displaystyle A+B=\{a+b:a\in A,b\in B\}}
of two subsets
A
{\displaystyle A}
and
B
{\displaystyle B}
of elements from an abelian group
G
{\displaystyle G}
, and the
h
{\displaystyle h}
-fold sumset of
A
{\displaystyle A}
,
h
A
=
A
+
⋯
+
A
⏟
h
{\displaystyle hA={\underset {h}{\underbrace {A+\cdots +A} }}\,}
.; Additive number theory has close ties to combinatorial number theory and the geometry of numbers."