TAILIEUCHUNG - Đề tài " Roth’s theorem in the primes "

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes. 1. Introduction Arguably the second most famous result of Klaus Roth is his 1953 upper bound [21] on r3 (N ), defined 17 years. | Annals of Mathematics Roth s theorem in the primes By Ben Green Annals of Mathematics 161 2005 1609 1636 Roth s theorem in the primes By Ben Green Abstract We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by giving a new proof of a rather more general result of Bourgain which because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis might be called a restriction theorem for the primes. 1. Introduction Arguably the second most famous result of Klaus Roth is his 1953 upper bound 21 on r3 N defined 17 years previously by Erdos and Turan to be the cardinality of the largest set A c N containing no nontrivial 3-term arithmetic progression 3AP . Roth was the first person to show that r3 N o N . In fact he proved the following quantitative version of this statement. Proposition Roth . r3 N N loglogN. There was no improvement on this bound for nearly 40 years until HeathBrown 15 and Szemeredi 22 proved that r3 N logN -c for some small positive constant c. Recently Bourgain 6 provided the best bound currently known. Proposition Bourgain . r3 N N loglogN logN 1 2. The author is supported by a Fellowship of Trinity College and for some of the period during which this work was carried out enjoyed the hospitality of Microsoft Research Redmond WA and the Alfred Renyi Institute of the Hungarian Academy of Sciences Budapest. He was supported by the Mathematics in Information Society project carried out by Renyi Institute in the framework of the European Community s Confirming the International Role of Community Research programme. 1610 BEN GREEN The methods of Heath-Brown Szemeredi and Bourgain may be regarded as highly nontrivial refinements of Roth s technique. There is a feeling that Proposition is close to the natural limit of this method.

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