section_path: Other subfields/Arithmetic combinatorics
section_name: Arithmetic combinatorics
paragraph_text: Arithmetic combinatorics
  
  If we begin from a fairly "thick" infinite set A {\displaystyle A}, does it contain many elements in arithmetic progression: a {\displaystyle a}, a + b, a + 2 b, a + 3 b, …, a + 10 b {\displaystyle a+b,a+2b,a+3b,\ldots,a+10b}, say? Should it be possible to write large integers as sums of elements of A {\displaystyle A} ?
  
  These questions are characteristic of arithmetic combinatorics. This is a presently coalescing field; it subsumes additive number theory (which concerns itself with certain very specific sets A {\displaystyle A} of arithmetic significance, such as the primes or the squares) and, arguably, some of the geometry of numbers, together with 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 A {\displaystyle A} being studied need not be sets of integers, but rather subsets of non-commutative groups, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of rings, in which case the growth of A + A {\displaystyle A+A} and A {\displaystyle A} · A {\displaystyle A} may be compared.