Arithmetic combinatorics

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

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]

Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured^[2] 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.

The Green–Tao theorem, proved by Ben Green and Terence Tao in 2004,^[3] 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.^[4] 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.

The Breuillard–Green–Tao theorem, proved by Emmanuel Breuillard, Ben Green, and Terence Tao in 2011,^[5] gives a complete classification of approximate groups. This result can be seen as a nonabelian version of Freiman's theorem, and a generalization of Gromov's theorem on groups of polynomial growth.

If A is a set of N integers, how large or small can the sumset

${\displaystyle A+A:=\{x+y:x,y\in A\},}$

the difference set

${\displaystyle A-A:=\{x-y:x,y\in A\},}$

and the product set

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

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.)

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

• Łaba, Izabella (2008). "From harmonic analysis to arithmetic combinatorics". Bull. Amer. Math. Soc. 45 (1): 77–115. doi:10.1090/S0273-0979-07-01189-5.
• Additive Combinatorics and Theoretical Computer Science Archived 2016-03-04 at the Wayback Machine, 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. arXiv:1108.3790. doi:10.1007/978-1-4614-6642-0_4. ISBN 978-1-4614-6642-0. S2CID 14979158.
• 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.
• 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.

• 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