Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive combinatorics refers to the special case when only the operations of addition and subtraction are involved.
For example: if A is a set of N integers, how large or small can the sumset
- ,
the difference set
- ,
and the product set
be, and how are the sizes of these sets related? (Not to be confused: the terms difference set and product set can have other meanings.)
The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.
Arithmetic combinatorics is explained in Green's review of "Additive Combinatorics" by Tao and Vu.
Famous quotes containing the word arithmetic:
“Under the dominion of an idea, which possesses the minds of multitudes, as civil freedom, or the religious sentiment, the power of persons are no longer subjects of calculation. A nation of men unanimously bent on freedom, or conquest, can easily confound the arithmetic of statists, and achieve extravagant actions, out of all proportion to their means; as, the Greeks, the Saracens, the Swiss, the Americans, and the French have done.”
—Ralph Waldo Emerson (1803–1882)