Any positive proportion of primes contains a 3-term arithmetic progression

Number theory 1 Minute

You need to know: Prime numbers, notation |A| for size of set A, arithmetic progression, limit superior \limsup.

Background: Let {\mathbb N} be the set of positive integers, and let {\cal P} be the set of primes. For n\in {\mathbb N}, let S_n=\{1,2,\dots,n\}, and let {\cal P}_n = {\cal P}\cap S_n be the set of primes not exceeding n. We say that subset A\subset N of integers has positive upper density if \limsup\limits_{n\to\infty}\frac{|A \cap S_n|}{n} > 0. Similarly, we say that subset A\subset {\cal P} of primes has positive upper density if \limsup\limits_{n\to\infty}\frac{|A \cap {\cal P}_n|}{|{\cal P}_n|} > 0.

The Theorem: On 25th February 2003, Ben Green submitted to arxiv a paper in which he proved that every subset of {\cal P} of positive upper density contains a 3-term arithmetic progression (3AP).

Short context: In 1939, Van der Corput proved that the set of primes contains infinitely many 3APs. In 1953, Roth proved that any subset of integers of positive upper density contains a 3AP. The Theorem provides a common generalisation of these two results. In a later work, Green and Tao proved that the same is true for arithmetic progressions of any length.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 5.5 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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