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:
“Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic.... It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.”
—Gottlob Frege (1848–1925)