additive number theory

English

Noun

additive number theory (uncountable)

1. (number theory) The 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 $A+B=\{a+b:a\in A,b\in B\}$ of two subsets $A$ and $B$ of elements from an abelian group $G$, and the $h$-fold sumset of $A$, $hA={\underset {h}{\underbrace {A+\cdots +A} }}\,$.

   Additive number theory has close ties to combinatorial number theory and the geometry of numbers.

   • 1966 [Macmillan], W. E. Deskins, Abstract Algebra, 1995, Dover, page 152,

     One of the famous theorems of additive number theory states that each positive integer is expressible in at least one way as the sum of the squares of not more than four positive integers.

   • 2006, Steven J. Miller, Ramin Takloo-Bighash, An Invitation to Modern Number Theory, Princeton University Press, page 303:

     The Circle Method is a beautiful idea for studying many problems in additive number theory. It originated in investigations by Hardy and Ramanujan [HR] on the partition function $P(n)$, the number of ways we can write $n$ as a sum of positive integers. Since then it has been used to study problems in additive number theory ranging from writing numbers as sums of primes or $k^{\mathsf {th}}$ powers (for fixed $k$) to trying to count how many twin primes there are less than $x$.

   • 2008, Krishnaswami Alladi, editor, Surveys in Number Theory, Springer, page ix:

     Additive number theory is a very broad area within which lies the theory of partitions, and the subject of representation of integers as sums of squares, cubes, etc. There is a vast literature in additive number theory on problems of the following type: Given a subset A of the integers, when does it form a basis for the integers? That is, when can every positive integer be expressed as the sum of at most a (fixed) bounded number of elements of A?

Translations

subfield of number theory

See also

• combinatorial number theory
• geometry of numbers
• multiplicative number theory

Further reading

• Combinatorial number theory on Wikipedia.Wikipedia
• Geometry of numbers on Wikipedia.Wikipedia