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Arithmetic combinatorics

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

Scope[edit]

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.


Ben Green explains arithmetic combinatorics in his review of "Additive Combinatorics" by Tao and Vu.[1]

Extensions[edit]

The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.[6]

Additive number theory

Approximate group

Corners theorem

Ergodic Ramsey theory

Problems involving arithmetic progressions

Schnirelmann density

Shapley–Folkman lemma

Sidon set

Sum-free set

Sum-product problem

(2008). "From harmonic analysis to arithmetic combinatorics". Bull. Amer. Math. Soc. 45 (1): 77–115. doi:10.1090/S0273-0979-07-01189-5.

Łaba, Izabella

Archived 2016-03-04 at the Wayback Machine, Luca Trevisan, SIGACT News, June 2009

Additive Combinatorics and Theoretical Computer Science

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. Vol. 43. New York: Springer Proceedings in Mathematics & Statistics. pp. 99–128. :1108.3790. doi:10.1007/978-1-4614-6642-0_4. ISBN 978-1-4614-6642-0. S2CID 14979158.

arXiv

E Croot, V Lev

Open problems in additive combinatorics

Terence Tao, AMS Notices March 2001

From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE

; Vu, Van H. (2006). Additive combinatorics. Cambridge Studies in Advanced Mathematics. Vol. 105. Cambridge: Cambridge University Press. ISBN 0-521-85386-9. MR 2289012. Zbl 1127.11002.

Tao, Terence

; 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.

Granville, Andrew

(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.

Mann, Henry

Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. . Vol. 164. New York: Springer-Verlag. ISBN 0-387-94656-X. MR 1395371.

Graduate Texts in Mathematics

Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. . Vol. 165. New York: Springer-Verlag. ISBN 0-387-94655-1. MR 1477155.

Graduate Texts in Mathematics

resources by Terence Tao

Some Highlights of Arithmetic Combinatorics

K Soundararajan

Additive Combinatorics: Winter 2007

Luca Trevisan

Earliest Connections of Additive Combinatorics and Computer Science