Additive number theory

Template:Short description 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. Principal objects of study include the sumset of two subsets Template:Mvar and Template:Mvar of elements from an abelian group Template:Mvar,

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

and the Template:Mvar-fold sumset of Template:Mvar,

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

Additive number theory

The field is principally devoted to consideration of direct problems over (typically) the integers, that is, determining the structure of Template:Math from the structure of Template:Mvar: for example, determining which elements can be represented as a sum from Template:Math, where Template:Mvar is a fixed subset. Two classical problems of this type are the Goldbach conjecture (which is the conjecture that Template:Math contains all even numbers greater than two, where Template:Math is the set of primes) and Waring's problem (which asks how large must Template:Mvar be to guarantee that Template:Math contains all positive integers, where

$A_k=\{0^k,1^k,2^k,3^k,\dots \}$

is the set of Template:Mvarth powers). Many of these problems are studied using the tools from the Hardy-Littlewood circle method and from sieve methods. For example, Vinogradov proved that every sufficiently large odd number is the sum of three primes, and so every sufficiently large even integer is the sum of four primes. Hilbert proved that, for every integer Template:Math, every non-negative integer is the sum of a bounded number of Template:Mvarth powers. In general, a set Template:Mvar of nonnegative integers is called a basis of order Template:Mvar if Template:Math contains all positive integers, and it is called an asymptotic basis if Template:Math contains all sufficiently large integers. Much current research in this area concerns properties of general asymptotic bases of finite order. For example, a set Template:Mvar is called a minimal asymptotic basis of order Template:Mvar if Template:Mvar is an asymptotic basis of order Template:Mvar but no proper subset of Template:Mvar is an asymptotic basis of order Template:Mvar. It has been proved that minimal asymptotic bases of order Template:Mvar exist for all Template:Mvar, and that there also exist asymptotic bases of order Template:Mvar that contain no minimal asymptotic bases of order Template:Mvar. Another question to be considered is how small can the number of representations of Template:Mvar as a sum of Template:Mvar elements in an asymptotic basis can be. This is the content of the Erdős–Turán conjecture on additive bases.

See also

• Shapley–Folkman lemma
• Additive combinatorics
• Multiplicative combinatorics
• Multiplicative number theory

References

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External links

• Template:Springer
• Template:Mathworld