Obstructions to uniformity, and arithmetic patterns in the primes

Details Obstructions to uniformity, and arithmetic patterns in the primes Obstructions to uniformity, and arithmetic patterns in the primes

2005-05-19

arxiv.org/abs/math/0505402v1

Mathematics

Number Theory

32 pages, no figures. Submitted, special edition of Quarterly J. Pure Appl. Math. in honour of John Coates

Scientific paper

In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving ``randomly'', and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.

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