Meaning of additive number theory | Babel Free

Definitions

The subfield of number theory concerning the behaviour of sumsets (integer sets added to each other elementwise).

uncountable

Examples

“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#43;B#61;#92;#123;a#43;b#58;a#92;inA,b#92;inB#92;#125; of two subsets A and B of elements from an abelian group G, and the h-fold sumset of A, hA#61;#92;underset#123;h#125;#123;#92;underbrace#123;A#43;#92;cdots#43;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.”

“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#92;mathsf#123;th#125; powers (for fixed k) to trying to count⟳ how many twin primes there are less than x.”

“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?”

CEFR level