Arithmetic Combinatorics

Arithmetic combinatorics is a branch of mathematics that arose at the intersection of number theory , combinatorics , ergodic theory and harmonic analysis .

Let be $\mathbb {N}$ ${\ displaystyle \ mathbb {N}}$ $\ mathbb {N}$ - many natural numbers, $\mathbb {E}$ ${\ displaystyle \ mathbb {E}}$ ${\ displaystyle \ mathbb {E}}$ - even $\mathbb {P}$ ${\ displaystyle \ mathbb {P}}$ ${\ mathbb P}$ - simple, and $\mathbb {S}$ ${\ displaystyle \ mathbb {S}}$ $\ mathbb S$ - the set of all squares of natural numbers.

The famous Lagrange theorem can be compactly formulated as the equality $\mathbb {S} +\mathbb {S} +\mathbb {S} +\mathbb {S} =\mathbb {N}$ ${\ displaystyle \ mathbb {S} + \ mathbb {S} + \ mathbb {S} + \ mathbb {S} = \ mathbb {N}}$ ${\ displaystyle \ mathbb {S} + \ mathbb {S} + \ mathbb {S} + \ mathbb {S} = \ mathbb {N}}$ , and no less famous Goldbach hypothesis - as $\mathbb {P} +\mathbb {P} \supseteq \mathbb {E}$ ${\ displaystyle \ mathbb {P} + \ mathbb {P} \ supseteq \ mathbb {E}}$ ${\ displaystyle \ mathbb {P} + \ mathbb {P} \ supseteq \ mathbb {E}}$ . Arithmetic combinatorics is engaged in studying the behavior of subsets of integers (as well as more complex algebraic structures) with respect to existing operations (in close collaboration with traditional number theory).

Additive combinatorics refers to a special case of arithmetic combinatorics, when only addition and subtraction operations are taken into account.

The sets under study can be subsets of algebraic structures other than integers, for example, groups , rings or fields . ^[one]

Arithmetic combinatorics is explained in Green's review of the book Additive Combinatorics by Tao and Wu .

Content

1 Example task
2 See also
3 notes
4 References

Example task

Let A be a set containing n integers, what will be the size of the set of sums

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

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

,

sets of differences (not to be confused with the difference of sets )

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

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

,

or sets of works (not to be confused with the work of sets )

A\times A:=\{xy:x,y\in A\}

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

How are the sizes of these sets related?

Notes

↑ A sum-product estimate in finite fields, and applications , Jean Bourgain, Nets Katz and Terence Tao, (2004), Geometric And Functional Analysis Volume 14, Number 1, 27-57, arxiv version

Links

Łaba, Izabella . From harmonic analysis to arithmetic combinatorics // Bull. Amer. Math. Soc. : journal. - 2008 .-- Vol. 45 , no. 01 . - P. 77-115 . - DOI : 10.1090 / S0273-0979-07-01189-5 .
Additive Combinatorics and Theoretical Computer Science , Luca Trevisan, SIGACT News, June 2009
Additive combinatorics with a view towards computer science and cryptography: An exposition , Khodakhast Bibak, 2012
Open problems in additive combinatorics , E Croot, V Lev
From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE , Tao, Terence , AMS Notices March 2001
Tao, Terence. Additive combinatorics / Terence Tao, Van H. Vu. - Cambridge: Cambridge University Press , 2006. - Vol. 105. - ISBN 0-521-85386-9 .
Additive Combinatorics. - American Mathematical Society , 2007. - Vol. 43. - ISBN 978-0-8218-4351-2 .
Mann, Henry. Addition Theorems: The Addition Theorems of Group Theory and Number Theory. - Corrected reprint of 1965 Wiley. - Huntington, New York: Robert E. Krieger Publishing Company , 1976. - ISBN 0-88275-418-1 .
Melvyn B. Nathanson. Additive Number Theory: the Classical Bases. - Springer-Verlag, 1996. - Vol. 164. - ISBN 0-387-94656-X .
Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and the Geometry of Sumsets. - Springer-Verlag, 1996. - Vol. 165. - ISBN 0-387-94655-1 .
Terence Tao , Some Highlights of Arithmetic Combinatorics
K Soundararajan, Additive Combinatorics: Winter 2007
Luca Trevisan, Earliest Connections of Additive Combinatorics and Computer Science

[1] A sum-product estimate in finite fields, and applications , Jean Bourgain, Nets Katz and Terence Tao, (2004), Geometric And Functional Analysis Volume 14, Number 1, 27-57, arxiv version

Arithmetic Combinatorics

Content

Example task

See also

Notes

Links

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