K-Poincaré algebra

In mathematics, arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis. It 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.

For 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

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

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.

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

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

Notes

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References

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• Additive Combinatorics and Theoretical Computer Science, Luca Trevisan, SIGACT News, June 2009
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 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
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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