Sumset Explained
In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets
and
of an
abelian group
(written additively) is defined to be the set of all sums of an element from
with an element from
. That is,
The
-fold iterated sumset of
is
where there are
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
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
is small (compared to the size of
); see for example
Freiman's theorem.
See also
References
. Henry Mann . Henry Mann . Addition Theorems: The Addition Theorems of Group Theory and Number Theory . Robert E. Krieger Publishing Company . Huntington, New York . 1976 . Corrected reprint of 1965 Wiley . 0-88275-418-1 .
- Book: Nathanson, Melvyn B. . 0722.11007 . Best possible results on the density of sumsets . 395–403 . Berndt . Bruce C. . Bruce C. Berndt . Diamond . Harold G. . Halberstam . Heini . Heini Halberstam . 3 . Hildebrand . Adolf . Analytic number theory. Proceedings of a conference in honor of Paul T. Bateman, held on April 25-27, 1989, at the University of Illinois, Urbana, IL (USA) . Progress in Mathematics . 85 . Boston . Birkhäuser . 1990 . 0-8176-3481-9 .
- Book: Nathanson, Melvyn B. . Additive Number Theory: Inverse Problems and the Geometry of Sumsets . 165 . . . 1996 . 0-387-94655-1 . 0859.11003 .
- Terence Tao and Van Vu, Additive Combinatorics, Cambridge University Press 2006.
External links
