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Guest column: additive combinatorics and theoretical computer science

Author:

Luca TrevisanAuthors Info & Claims

ACM SIGACT News, Volume 40, Issue 2

Pages 50 - 66

https://doi.org/10.1145/1556154.1556170

Published: 20 June 2009 Publication History

Abstract

Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties and patterns that can be expressed via additions and multiplications.

In the past ten years, additive and arithmetic combinatorics have been extremely successful areas of mathematics, featuring a convergence of techniques from graph theory, analysis and ergodic theory. They have helped prove long-standing open questions in additive number theory, and they offer much promise of future progress.

Techniques from additive and arithmetic combinatorics have found several applications in computer science too, to property testing, pseudorandomness, PCP constructions, lower bounds, and extractor constructions. Typically, whenever a technique from additive or arithmetic combinatorics becomes understood by computer scientists, it finds some application.

Considering that there is still a lot of additive and arithmetic combinatorics that computer scientists do not understand (and, the field being very active, even more will be developed in the near future), there seems to be much potential for future connections and applications.

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Cited By

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Guest column: additive combinatorics and theoretical computer science

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Published In

cover image ACM SIGACT News

ACM SIGACT News Volume 40, Issue 2

June 2009

91 pages

ISSN:0163-5700

DOI:10.1145/1556154

Issue’s Table of Contents

Copyright © 2009 Author.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 20 June 2009

Published in SIGACT Volume 40, Issue 2

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Cited By

Jetchev DPietrzak K(2014)How to Fake Auxiliary InputTheory of Cryptography10.1007/978-3-642-54242-8_24(566-590)Online publication date: 2014
https://doi.org/10.1007/978-3-642-54242-8_24
Bibak K(2013)Additive Combinatorics: With a View Towards Computer Science and Cryptography—An ExpositionNumber Theory and Related Fields10.1007/978-1-4614-6642-0_4(99-128)Online publication date: 5-Apr-2013
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https://doi.org/10.1002/rsa.20496
Kohayakawa YLee SRödl VRandall D(2011)The maximum size of a Sidon set contained in a sparse random set of integersProceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms10.5555/2133036.2133050(159-171)Online publication date: 23-Jan-2011
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Haramaty EShpilka AMitzenmacher MSchulman L(2010)On the structure of cubic and quartic polynomialsProceedings of the forty-second ACM symposium on Theory of computing10.1145/1806689.1806736(331-340)Online publication date: 5-Jun-2010
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