Editing Number theory (section)

=== Arithmetic combinatorics and additive number theory ===
{{Main|Arithmetic combinatorics|Additive number theory}}

Combinatorics in number theory starts with questions like the following: Does a fairly "thick" [[infinite set]] <math>A</math> contain many elements in arithmetic progression: <math>a</math>,
<math>a+b, a+2 b, a+3 b, \ldots, a+10b</math>? Should it be possible to write large integers as sums of elements of <math>A</math>?

These questions are characteristic of ''arithmetic combinatorics,'' a coalescing field that subsumes [[additive number theory]] (which concerns itself with certain very specific sets <math>A</math> of arithmetic significance, such as the primes or the squares), some of the [[geometry of numbers]], as well as some rapidly developing new material. Its focus on issues of growth and distribution accounts in part for its developing links with [[ergodic theory]], [[finite group theory]], [[model theory]], and other fields.

The term ''additive combinatorics'' is also used; however, the sets <math>A</math> being studied need not be sets of integers, but rather subsets of non-commutative [[Group (mathematics)|groups]], for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of [[ring (mathematics)|rings]], in which case the growth of <math>A+A</math> and <math>A</math>·<math>A</math> may be compared.