Arithmetic combinatorics

In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.

Scope

Arithmetic combinatorics is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive combinatorics is the special case when only the operations of addition and subtraction are involved.

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

Important results

Szemerédi's theorem

Main article: Szemerédi's theorem

Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem.

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

$A\cdot A:=\{xy:x,y\in A\}$

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.)

Extensions

The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.

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

• Łaba, Izabella (2008). "From harmonic analysis to arithmetic combinatorics". Bull. Amer. Math. Soc. 45 (01): 77–115. doi:10.1090/S0273-0979-07-01189-5.
• Additive Combinatorics and Theoretical Computer Science, Luca Trevisan, SIGACT News, June 2009
• Bibak, Khodakhast (2013). "Additive combinatorics with a view towards computer science and cryptography". In Borwein, Jonathan M.; Shparlinski, Igor E.; Zudilin, Wadim (eds.). Number Theory and Related Fields: In Memory of Alf van der Poorten. Springer Proceedings in Mathematics & Statistics, Vol. 43, Springer, New York. pp. 99–128. doi:10.1007/978-1-4614-6642-0_4. ISBN 978-1-4614-6642-0.
• 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
• Tao, Terence; Vu, Van H. (2006). Additive combinatorics. Cambridge Studies in Advanced Mathematics. Vol. 105. Cambridge: Cambridge University Press. ISBN 0-521-85386-9. Zbl 1127.11002.
• Granville, Andrew; Nathanson, Melvyn B.; Solymosi, József, eds. (2007). Additive Combinatorics. CRM Proceedings & Lecture Notes. Vol. 43. American Mathematical Society. ISBN 978-0-8218-4351-2. Zbl 1124.11003.
• Mann, Henry (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. {{cite book}}: External link in |publisher= (help)
• Melvyn B. Nathanson (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. Vol. 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. Vol. 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