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Generalising the HardyLittlewood method for primes
 In: Proceedings of the international congress of mathematicians
, 2007
"... Abstract. 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 ..."
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Abstract. 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
"... Abstract. We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part ..."
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Abstract. We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part of classical Fourier analysis that investigates the relationship between geometric and Fourieranalytic properties of singular measures. It became clear over the years that the theory would have to involve sophisticated geometric and combinatorial input. Two particularly important turning points were Fefferman’s work in the 1970s invoking the ”Kakeya problem ” in this context, and Bourgain’s application of Gowers’s additive number theory techniques to the Kakeya problem almost 30 years later. All this led harmonic analysts to explore areas previously foreign to them, such as combinatorial geometry, graph theory, and additive number theory. Although the Kakeya and restriction problems remain stubbornly open, the