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# The dichotomy between structure and randomness , arithmetic progressions , and the primes

@inproceedings{Tao2007TheDB, title={The dichotomy between structure and randomness , arithmetic progressions , and the primes}, author={Terence Tao}, year={2007} }

- Published 2007

A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of… CONTINUE READING

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