Arithmetic combinatorics

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.

For example: if A is a set of N integers, how large or small can

the sumset A + A := { x + y : x,y ∈ A },

the difference set A − A := { x − y : x,y ∈ A },

and the product set A × A := { xy : x,y ∈ A }

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 sets other than the integers. e.g. groups and rings.

[edit] See also

• Green–Tao theorem
• Szemerédi's theorem
• Ergodic Ramsey theory
• Additive number theory
• Problems involving arithmetic progressions

[edit] Further reading

• Some Highlights of Arithmetic Combinatorics
• FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS

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