Arithmetic combinatorics
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Arithmetic combinatorics is concered with combinatorial estimates associated with arithmetic operations such as addition, subtraction, and multiplication. A typical question is the following: if A is a set of N integers, how large or small can the sumset A+A := { x + y: x,y in A }, the difference set A-A := { x - y: x,y in A }, and the product set A*A := { xy: x,y in A } be, and how are the sizes of these sets related? A typical result is the following: if |A+A| <= CN, then |A+A+A| <= C^3 N.
Not to be confused: the terms difference set and product set can have other meanings.
The most developed part of arithmetic combinatorics is additive combinatorics.
See also
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