 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
 ,
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 belong to other spaces than the integers. e.g. groups, rings and fields.^{[1]}
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
 Sumfree set
 Szemerédi's theorem
References
 ^ A sumproduct estimate in finite fields, and applications, Jean Bourgain, Nets Katz and Terence Tao, (2004), Geometric And Functional Analysis Volume 14, Number 1, 2757, arxiv version
 Izabella Laba (2008). "From harmonic analysis to arithmetic combinatorics" (PDF). Bull. Amer. Math. Soc. 45 (01): 77–115. doi:10.1090/S0273097907011895. http://www.ams.org/bull/20084501/S0273097907011895/S0273097907011895.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 0882754181.
 Melvyn B. Nathanson (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. 164. SpringerVerlag. ISBN 038794656X.
 Melvyn B. Nathanson (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. 165. SpringerVerlag. ISBN 0387946551.
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
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