The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view

By Bryna Kra. 2005

58 citations recorded by the source corpus.

A long standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ec7c1b3f85852d5d5e4f61cfb836bb341fca05a8

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma (2006, background): “…crucial in the recent result [13] that the primes contained arbitrarily long progressions; this argument was almost entirely ﬁnitary in nature, yet at the same time it relied heavily on ideas from the inﬁnitary world of ergodic theory (see [17], [15] for further discussion of this connection).”
Some open problems on multiple ergodic averages (2011, background): “[130] B.”
The dichotomy between structure and randomness, arithmetic progressions, and the primes (2005, result): “As mentioned in the introduction, these results are discussed in other surveys [42], [25], [55], [56], [37], and we will only sketch some highlights here.”