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 (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

$A+A := \{x+y: x,y \in A\}$,

the difference set

$A-A := \{x-y: x,y \in A\}$,

and the product set

$A\times A := \{xy: x,y \in A\}$

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 belong to other spaces than the integers. e.g. groups, rings and fields.

Arithmetic combinatorics is explained in Green's review of "Additive Combinatorics" by Tao and Vu.

See also

• Additive number theory
• Corners theorem
• Ergodic Ramsey theory
• Green–Tao theorem
• Problems involving arithmetic progressions
• Schnirelmann density
• Shapley–Folkman lemma
• Sidon set
• Sum-free set
• Szemerédi's theorem

References

• Izabella Laba (2008). "From harmonic analysis to arithmetic combinatorics" (PDF). Bull. Amer. Math. Soc. 45 (01): 77–115. doi:10.1090/S0273-0979-07-01189-5. http://www.ams.org/bull/2008-45-01/S0273-0979-07-01189-5/S0273-0979-07-01189-5.pdf. 
• Additive Combinatorics and Theoretical Computer Science, Luca Trevisan, SIGACT News, June 2009
• Open problems in additive combinatorics, E Croot, V Lev
• From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE, Terence Tao, AMS Notices March 2001
• Terence Tao, Van Vu (2006). Additive Combinatorics. Cambridge University Press. ISBN 9780521853866. 
• Andrew Granville, Melvyn B. Nathanson, Jozsef Solymosi, editors. (2006). Andrew Granville, Melvyn Bernard Nathanson, Jozsef Solymosi. ed. Additive Combinatorics. AMS Bookstore. ISBN 9780821843512. 
• Henry Mann (1976). Addition Theorems: The Addition Theorems of Group Theory and Number Theory (Corrected reprint of 1965 Wiley ed.). Huntington, New York: Robert E. Krieger Publishing Company. ISBN 0-88275-418-1. 
• Melvyn B. Nathanson (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. 164. Springer-Verlag. ISBN 0-387-94656-X. 
• Melvyn B. Nathanson (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. 165. Springer-Verlag. ISBN 0-387-94655-1. 

Further reading

• Some Highlights of Arithmetic Combinatorics, resources by Terence Tao
• Additive Combinatorics: Winter 2007, K Soundararajan
• Earliest Connections of Additive Combinatorics and Computer Science, Luca Trevisan