ar X iv : 0 81 1 . 31 03 v 1 [ m at h . C O ] 1 9 N ov 2 00 8 DECOMPOSITIONS , APPROXIMATE STRUCTURE , TRANSFERENCE , AND THE HAHN - BANACH THEOREM
نویسنده
چکیده
We discuss three major classes of theorems in additive and extremal com-binatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finite-dimensional Hahn-Banach theorem can be used to give short and transparent proofs of many results of these kinds. Amongst the applications of this method is a much shorter proof of one of the major steps in the proof of Green and Tao that the primes contain arbitrarily long arithmetic progressions. In order to explain the role of this step, we include a brief description of the rest of their argument. A similar proof has been discovered independently by Reingold, Trevisan, Tulsiani and Vadhan [RTTV].
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