2007
DOI: 10.1090/s0273-0979-07-01189-5
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From harmonic analysis to arithmetic combinatorics

Abstract: Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably best-known result, and the one that brought it to global prominence, is the proof by Ben Green and Terence Tao of the long-standing conjecture that primes contain arbitrarily long arithmetic progressions. There are many accounts and expositions of the Green-Tao theorem, including the articles by Kra [119] and Tao [182]… Show more

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Cited by 23 publications
(18 citation statements)
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“…The Nikodym conjecture is a close cousin of the more well-known Kakeya conjecture. The connection between these geometrical questions and problems in Fourier analysis and PDE is described in [Laba08] and [Tao01].…”
Section: Examples Of the Polynomial Methodsmentioning
confidence: 99%
“…The Nikodym conjecture is a close cousin of the more well-known Kakeya conjecture. The connection between these geometrical questions and problems in Fourier analysis and PDE is described in [Laba08] and [Tao01].…”
Section: Examples Of the Polynomial Methodsmentioning
confidence: 99%
“…What happens X ·X contains unusually few elements; for example what if |X · X| ≤ 2|X|? The structure of such sets is studied in additive combinatorics, which is a burgeoning area of mathematics with applications in computer science [35], harmonic analysis [25], number theory [28], combinatorics and elsewhere. It has spanned a host of new techniques (e.g.…”
Section: 2mentioning
confidence: 99%
“…The structure of incidences between points and various geometric objects is of central importance in discrete geometry, and theorems that elucidate this structure have had applications to, for example, problems from discrete and computational geometry [18,23], additive combinatorics [8,13], harmonic analysis [17,22], and computer science [12]. The study of incidence theorems for finite geometry is an active area of researche.g.…”
Section: Introductionmentioning
confidence: 99%