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Generalising the HardyLittlewood method for primes
 IN: PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2007
"... The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number o ..."
Abstract

Cited by 9 (2 self)
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The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the HardyLittlewood method has been generalised to obtain, for example, an asymptotic for the number of 4term arithmetic progressions of primes less than N.
FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS
, 2008
"... 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 bestknown result, and the one that brought it to global prominence, is the proof by Ben Green and Terence ..."
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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 bestknown result, and the one that brought it to global prominence, is the proof by Ben Green and Terence Tao of the longstanding conjecture that primes contain arbitrarily long arithmetic progressions. There are many accounts and expositions of the GreenTao theorem, including the articles by Kra [119] and Tao [182] in the Bulletin. The purpose of the present article is to survey a broader, highly interconnected network of questions and results, built over the decades and spanning several areas of mathematics, of which the GreenTao theorem is a famous descendant. An old geometric problem lies at the heart of key conjectures in harmonic analysis. A major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles. An unexpected argument points harmonic analysts towards additive number theory, with consequences that could have hardly been anticipated. We will not