Carl Neumann

In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets A and B of an abelian group G (written additively) is defined to be the set of all sums of an element from A with an element from B. That is,

$A+B=\{a+b:a\in A,b\in B\}.$

The n-fold iterated sumset of A is

$nA=A+\cdots +A,$

where there are n summands.

Many of the questions and results of additive combinatorics and additive number theory can be phrased in terms of sumsets. For example, Lagrange's four-square theorem can be written succinctly in the form

$4◻=\mathbb{\mathbb{N}},$

where $◻$ is the set of square numbers. A subject that has received a fair amount of study is that of sets with small doubling, where the size of the set A + A is small (compared to the size of A); see for example Freiman's theorem.

See also

• Minkowski addition (geometry)
• Restricted sumset
• Sidon set
• Sum-free set
• Schnirelmann density
• Shapley–Folkman lemma

References

• 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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• 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
• Terence Tao and Van Vu, Additive Combinatorics, Cambridge University Press 2006.