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

Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured^[1] 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.

Green-Tao theorem and extensions

The Green–Tao theorem, proved by Ben Green and Terence Tao in 2004,^[2] states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words there exist arithmetic progressions of primes, with k terms, where k can be any natural number. The proof is an extension of Szemerédi's theorem.

In 2006, Terence Tao and Tamar Ziegler extended the result to cover polynomial progressions.^[3] More precisely, given any integer-valued polynomials P₁,..., P_k in one unknown m all with constant term 0, there are infinitely many integers x, m such that x + P₁(m), ..., x + P_k(m) are simultaneously prime. The special case when the polynomials are m, 2m, ..., km implies the previous result that there are length k arithmetic progressions of primes.

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.^[4]

Notes

^ Erdős, Paul; Turán, Paul (1936). "On some sequences of integers" (PDF). Journal of the London Mathematical Society. 11 (4): 261–264. doi:10.1112/jlms/s1-11.4.261. MR 1574918..
^ Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics. 167 (2): 481–547. arXiv:math.NT/0404188. doi:10.4007/annals.2008.167.481. MR 2415379..
^ Tao, Terence; Ziegler, Tamar (2008). "The primes contain arbitrarily long polynomial progressions". Acta Mathematica. 201 (2): 213–305. arXiv:math.NT/0610050. doi:10.1007/s11511-008-0032-5. MR 2461509..
^ Bourgain, Jean; Katz, Nets; Tao, Terence (2004). "A sum-product estimate in finite fields, and applications". Geometric and Functional Analysis. 14 (1): 27–57. doi:10.1007/s00039-004-0451-1. MR 2053599.

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. Vol. 43. New York: Springer Proceedings in Mathematics & Statistics. 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. MR 2289012. 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)
Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. Vol. 164. New York: Springer-Verlag. ISBN 0-387-94656-X. MR 1395371.
Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. New York: Springer-Verlag. ISBN 0-387-94655-1. MR 1477155.