additive combinatorics

English

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additive combinatorics

Etymology

Coined circa early 2000s by Australian-American mathematician Terence Tao for a rapidly developing field growing out of combinatorial number theory, named differently to reflect a changed emphasis in the problems being studied.^[1]^[2]

Noun

additive combinatorics (uncountable)

(mathematics) A subbranch of combinatorics that concerns additive problems expressed using sumsets.
One major area of study in additive combinatorics is that of inverse problems: for instance, given the sumset $A+B$ is small in size, what can we say about the structures of $A$ and $B$ ? In the case of integer sumsets, Freiman's theorem provides a partial answer.
- 2007, Andrew Granville, Additive Combinatorics, American Mathematical Society, [1].
- 2014, Terence Tao, Hilbert's Fifth Problem and Related Topics, American Mathematical Society, page 14:
  We now discuss what appears at first glance to be an unrelated topic, namely that of additive combinatorics (and its noncommutative counterpart, multiplicative combinatorics). One of the main objects of study in either additive or multiplicative combinatorics are approximate groups — sets $A$ (typically finite) contained in an additive or multiplicative ambient group $G$ that are "almost groups" in the sense that they are "almost" closed under either addition or multiplication.
- 2016, Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, Taylor & Francis (CRC Press), page xi:
  This book deals with additive combinatorics, a vibrant area of current mathematical research. Additive combinatorics—an offspring of combinatorial number theory and additive number theory—can be described as the study of combinatorial properties of sumsets (collections of sums with terms from given subsets) in additive structures.

Synonyms

(discipline that concerns combinatorics problems expressed using sumsets): combinatorial number theory

Translations

subbranch of combinatorics

References

^ 2009 July, Ben Green, Book Reviews: Additive combinatorics, by Terence C. Tao and Van H. Vu, Bulletin of the American Mathematical Society, New Series, Volume 46, Number 3, pages 489–497—The term additive combinatorics was coined a few years ago by Terry Tao to describe a rapidly developing and rather exciting area of mathematics.
^ 2009, Imre Ruzsa, Part II: Sumsets and Structure, Alfred Geroldinger, Imre Ruzsa (editors), Combinatorial Number Theory and Additive Group Theory, Springer (Birkhäuser), CRM, page 88—The course is devoted to several aspects of combinatorial number theory, or additive combinatorics as it is now often called. This change in terminology reflects a shift in the emphasis of problems investigated.